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Maps, plates, charts, etc., mey be filmed et different reduction retioa. Those too large to be entirely included in one exposure ara filmed beginning in the upper left hand comer, left to right and top to bottom, as many frames es required. The following diagrama illustrate the method: Les certes, planches, tableaux, etc., peuvent Atre filmAs A des taux de rMuction diffArents. Lorsque le document est trop grond pour Atre reproduit en un seui ciichA, 11 est f ilm6 A partir de I'engle sup^rieur gauche, de gauche A droite, et de heut en bas, an prenant le nombre d'imagas nteessaire. Les diagrammes suivants illustrent la mtthoda. 1 2 3 1 2 3 4 5 6 4 m^ aIr :'t SmPLE EXERCISES nr MEE"STJK ATIOlSr ; DESIGNED FOR THB CSH OP CANADIAN COMMON AND GRAMMAR SCHOOLS JOHN HERBERT SANGSTER, M.A., M.D., HIAD HASTBR NORUIL SOBOOL VOR ONTARIO. ^ONTREAL : PRINTED AND PUBLISHED BY JOHN LOVELL, AND FOR SALS AT THE BOOKSTORBg. 1872. Entered, according to the Act of the Provincial Parliament, in the year one thousand eight hundred and sixty-seven, by John Lovell, in the OflBce of the Registrar of the Dominion of Canada. ^i PREFACE. ^ ^i This little book is not intended to supersede the more elaborate text-books upon the same subject, in use in our Schools, but rather to serve as an introduction to one or other of them. The great mass of Common and Grammar School pupils have not time, amid the many o^er import- ant studies claiming their attention, to devote to any lengthened course of instruction upon Mensuration. All that the teacher can ordinarily hope, under existing cir- cumstances, to accomplish in this department, is to make his scholars capable of readily computing the area of regular surfaces and the volume or capacity of regular solids. Where more is attempted, it is, as a general thing, done at the expense of other important branches of instruction. Those who are intended for professions which require an intimate knowledge of Land Surveying, Astronomy, Gauging, &c., may, of course, profitably devote one or more entire years to the study of the various departments of mensuration, but for general purposes — for the farmer, the mechanic, the merchant, a knowledge of the mensuration of ordinary surfaces and solids is amply sufficient, and it is for such that the following pages have been thrown together. The rules are given in the form of formulas, because it is believed that they are thus much more readily and last- ingly remembered, and a very little effort on the part of the teacher will enable the pupil both to understand the depend- ence of the rules upon one another, and the interpretation and application of the formulas. Toronto, October, 1867. w CONTENTS. ''''^ ■'■■■■ ' ,' ' \, . * PA«E. l)efinitions 7 Formulas for Mensuration of Surfaces 17 Formulas for "M ensuration of Solids 20 Illustrations and Exercises 23 Miscellaneous Exercises fl4 Problems for Practice... 65 Tables 72 ' V (•• MENSURATION. DEFINITIONS. '77t« teacher is expected to draw on the hlacJchoard or slate, ^gures illustrating those definitions. 1 A figure is that which is enclosed by one or more boundaries. 2. \ plane figure is enclosed by one or more lines, which lie on the same plane or flat surface. 3. A solid body is that which is contained or bounded by one or more surfaces. 4.' A plane figure or surfaced said to have two dimensions, viz : length and breadth ; a solid is said to haVe three dimensions, viz : length, breadth and thickness. 5. The area of a plane figure is the number of square units of measurement contained within its bounding line or lines ; the volume of a solid body is the number of cubic units of measurement contained within its bounding surface or surfaces. 6. Mensuration consists in the determination of the areas of surfaces and the volume of solids from their linear dimensions. 7. A plane rectilineal angle is the mutual inclination of two straight lines towards one another — which meet but are not in the same straight line. Note.— The magaitude of the angle depends upon tU« rate of dirsrgence of the lines — not upon their length. 8 IfENSURATIOK. i • I 8. When one straight line, standing upon another, makes the adjacent angles equal, each of them is called a rigJit angle; and the line which stands upon the other is called a perpendicular to it. 9. An angle less than a right angle is called an acute angle ; an angle greater than a right angle is called an ohtune angle. 10. Parallel straight lines are those that lie in the same plane and which have the same direction, so that being produced ever so far, both ways, they never meet. 11. A triangle is a figure contained by three straight lines. 12. An equilateral ^n a wjf?^ has all of its sides equal; an isosceles triangle has two of its sides equal ; and a scalene triangle has all of its sides unequal. 13. "A right-angled triangle has one of its angles a right angle ; an obtuse-angled triangle has one of its angles an obtuse angle ; and an acute-angled triangle has all three of its angles acute angles. The two latter are often c&Wed oblique-angled triangles. 14. A quadrilateral figure is that which is enclosed by four straight lines. 15. A trapezium is a quadrilateral figure having no two of its sides parallel. 16. A trapezoid is a quadrilateral figure having one pair of opposite sides parallel. 17. A parallelogram is a quadrilateral figure having each pair of opposite sides parallel. 18. A rectangle or oblong is a parallelogram whose angles are all right angles, but its adjacent sides are not equal, t. c, its length is greater than its breadth. 19. A square is a parallelogram whose angles are all right angles and its sides are all equal. DE7INITI0NI. 9 angles ; equal, Fig. 1— Pentagon. 20.^ Thn diagonal of a quadrilateral figure is a straight line joining its opposite angles. 21. A. polygon or multilateral Jigure is a figure contained by more than four straight lines. 22. A regular polygon is one whose sides are all equal to one another, as also are its angles. 23. Polygons are named from the number of their sides — thus a five-sided, polygon is called a pentagon ; a six- sides polygon, is called a hexagon; a seven-sided polygon is called a : heptagon ; an eight-sided polygon is called an octagon, &c. 24. The apothcm of a regular polygon is a perpendicular from its centre on any of its sides ; as AB, Fig. 1 or 2. 25. A circle is a plane figure, bounded by one line, called the circumference, j^ and is such that every part of the Fig. 2— Hexagon. circumference is equally distant from a point within, called the centre. NoTB.— The circumference is the bounding line— the circle the space inclosed. 26. The diameter of a circle is a straight line passing through the centre, and terminated both ways in the circumference. 27. A semicircle is the figure contained by the diameter and the part of the circumference cut off by the dia- meter. 28. The radius of a circle is half the diameter, or is a straight line joining the centre of the circle with th« circumference. 10 MENSURATION. !i 29. An arc of a circle is any part of the circumference. 30. A c}iovd of a circle is any straight line joining the extremities of an are. - ^ 31. A segment of a circle is the figure contained by a chord and the arc of the circumference cut of by the chord. 32. A sector of a circle is the figure contained by an arc of the circumference, and the two radii joining its extremities with the centre of the circle. 33. A lune is the figure contained between the circular arcs of two dissimilar circular segments which have a com- mon chord. Fig. 3. O „ Thus in Fig. 3 AB is a chord, ACB and ADB are two dissimilar circular segments, ACBD is a lune. 34. A degree is the 360th part of the circumference oi a circle. Note.— The length of the degree depends upon the magnitude of the circle, 35. Concentric circles are such as have a common centre. A circular annulus is the figure inclosed between the circumferences of two concentric circles. The '£^rimeter or periphery of any figure is its circum- ference, or the aggregate length of all its boundaries. 38. A polyhedron is any solid contained by planes, which planes are called its sieve's or faces. The lines boundino* its sides arc called its edgus. 36 37. DEITNITIONS. 11 39. A regular poli/liedron is one whose sides are equal and regular figures of tho same kind, and whose solid angles are equal. 40. There arc only five regular polyhedrons, viz. :— The tetrahedron contained by four equilateral tri- angles. Fig. 4. The hexaJiedron contained by six squares. Fig. 5. The octahedron contained by eight equilateral tri- angles. Fig. 6. The dodecahedron contained by twelve pentagons. Fig. 7. The icosahedron contained by twenty equilateral tri- angles. Fig. 8. Fig. 4. Fig. r,. . ^ '1 Fig. 6. Fig. 7. 12 MENSURATION. :?'■■-: Fig. «. 41. A prism is a solid contained by plane figures, of which two are equal, similar and opposite ; with their sides parallel each to each, and the other sides are parallelo- grai^s. 42. The ends or terminating planes of the prism are the two similar sides, and the edges of these are called terminating edges to dis- tinguish them from the lateral A Polygonal Right sides and edges. The prism is^^s™? ^^^^^ ^^^ , °, J ^ FGHIK its ends. AB, tnangular,rectangular,square or ^^^ ^^^^ &c., terminat- j>olggonal, according as its termi- ing edges. AF, GB, OH, nating planes or ends are tri- &c., its lateral edgeg. angles, rectangles, squares or polygons. When the lateral edges are perpendicular to the end, the prism is called a rigJtt prism, when otherwise, an oblique prism. The line joining the centres of the terminating planes of a prism is called its axis. v 43. A parallelopiped is a prism having parallelograms for its terminating planes or ends. 44. A aihe is a solid contained by six equal squares. DEFINITIONS. Id 45. A pyramid is a solid having any rectilineal figure for its base ; and for its other sides triangles, which have a common vertex. The pyramid is triangular, square, rectangular, &c., according as its base is a triangle, a square, a rectangle,' &c. 46. When the base is a regular figure, -^ a line joining its centre with the vertex or the pyramid is called the axis of the pyramid. When ^ K«S"^»'' Pyra™w .V , , ABODE its base. BCS, the axis is at right angles to the acs, &c., its sides, base, the pyramid is called a regular pyramid. 47. A cone is a round pyramid having Fig- li. a circle for its base, and is con- ceived to be produced by the re- volution of a right-angled triangle about its perpendicular side which ' remains fixed. The line joining the vertex of the cone with the centre of the base is called the axis of the cone. Fig. 11. 48. A right cone is one in which the axis is perpendicular to the base— all other' cones are called oblique. 49. A cylinder is a prism having circles for its ends or termi- nating planes, and is conceived to be produced by the revolu- tion of a rectangle about one of its sides, which remains fiz«d. Fig. 12. ■I-, Fig- 12. 14 MENSURATION. ! 4 »■'. I ^Q. Asj^here OT gloheis&BoMhody - F*«18. which may be supposed to be produced by the revolution of a ^ semi-circle about its diameter which remains fixed. Fig, 13. 51. A segment of a sphere is a part G of it cut off by a plane ; a seg- ment of a pyramid, cone, cylin- der or other solid, with a plane base is a portion cut off from the top by a plane parallel to the base. 52. A fnistum of a solid is the portion Fig. 14, contained between the base and a plane parallel to the base as in fig. 14 ; the frustum or zone of a sphere is the portion cut off by two parallel^ phnes as ADGH in Fig. 13. 53. An ellipse or oval (Fig. 15) is a plane figure bounded by a curved line such that the sum of the • ^ distances of any point in its^j circumference from two given points in it is constant, i. e., is equal to a given straight line. Thus Kig. 15 atbcl is'an ellipse because i^'^pf is constant ; /and f are the foci, c the centre, td the transverse and ah the conjugate diameter or axis, sm is an ordinate and six a double ordinate, tm and md are the abscisses to tlio ordinate sm, 54. The two given points are called the fod of the ellipse, and the middle of the line joining them is called the centre of the ellipse. The distance of either focus from the centre is called the eccentridty of the ellipse. 55. The major or long axis or transverse diameter of an ellipse, is a line through both foci, and terminating in the bounding curve. DEMKlTiONS. 16 56. The minor or short axis or conjugate diameter, is a line passing through the centre, at right angles to the major axis, and terminating hoth ways in the bounding curve. 57. An ordinate to either axis is a line drawn from any point in the curve perpendicular to the axis ; when it is continued to meet the curve on the other side, it is called a double ordinate. 58. Each of the segments into which the ordinate divides the axis is called an absciss. 59. A parabola is a curve such that any point of it is equally distant from a given point within the curve, and a given line without it. Thus if the curve mvn is such that any point JO in it is equally distant .from the point /, and the line ab, tliat is, if pa is equal topf, than the curve mvn is a para- "*" fooln. Also/ is the focua ; po is the ordinate, and pe the double ordinate ot boat V is the vertex, and ov is the absciss. The double ordinate through /is called the parameter. 60. An hyperbola is a curve, such that the difference between the distances of any point in it from two given points, one within, and the other without the curve, is equal to a given line. Thus it any point p in the curve pbe is such that pr-pf='ab, a gireM jin«, then the ourvs pbe i» an hyperbola. Alao/and/* ar« Vanfoei, oft is id MENSURATIOir. li! til •J the ^ranflrera« axh, eis the ««n/re ,- mn is ihe conjtif^ate axis, the points m and n being distant from o or 6 by cf or <;/*, <. e., by the eccentricity ; po is the ordinate ; and pe the double ordinate or base ; ab is the smaller tUfacia$ ; and oa the greater a6«ct<«. 61 . A paraboloid or parabolic conoid is a solid generated by the revolution of a parabola about its axis, which remains fixed. NoTB.— A fruBtnm of a paraboloid -is a portion contained bet «reen twr parallel planes perpendicular to its axis. 62. A spheroid is a solid generated by the revolution of an ellipse about one of its axis which remains fixed. 63: A spheroid is said to be oblate or prolate, according as it is the conjugate or the transverse axis that is fixed. KOTB. -The fixedaxis is called the polar axis, and the rerolving axis the equatorial axis. 64. A segmmt of a spheroid is a portion cut off by a plane perpendicular to one of its axes. Note.— When the plane is perpendicular to the fixed axis, the base is a flirde, and the i^egment is said to be circular; when the plane iit perpendi- cular to the revolving axis, the segment is called an elliptical one, because the base is an ellipse. 65. The middle zone of a spheroid or of a sphere is a portion contained between two parallel planes perpen- dicular to an axis and equally distant from the centre. 66. An hyperboloid or hyperbolic conoid is a solid gene- rated by an hyperbola about its axis which remains fixed. NoTB— A firustum of an hyperboloid is a portion of it contained between two parallel planes perpendicular to its axis. >>'"» ■■ t;;,^-', ^ * ^ ■ '■ 8URFACBS. ■ / ' ' '■^■17 ; MENSURATION OF SURFACES. * SYNOPSIS OP FORMULAS. Let A ==■ area, h = base, jp = perpendicular or altitude, d =^ diagonal. Square. A = 6' (i) /. h = ^A (ii). Also A - ^cP (in) .*. d='J2A(iy). A Rectangle or PARALLELoaRAM. A-=^hp (v) :.b=- (vi) andj3 = -=-(vii). Rectangle. A = h\l{d-th) {d-h) (viii). Right-Angled Triangle. Let 6 = the hypotbenuse, 'p' = the perpendicular from the right angle on the hypotbenuse, and s and s' = the segments into which this divides the bypothenuse, s being that adjacent to the base of the triangle. Then h = ^b^+p^ (ix) ; 6 = VA^^ (X) ; p = ^h^'^n? (XI) ; s =-|'(xil) ; s' = -t(xiii) ; and p' = Vss^ (xiv). Triangle. A = ^hp (xv); 2A 6 = — (xvi) : and p ~ — (xvii). Also, if a, 6, c be the three sides and A- = J(a + fe + c) then A = Vs(« - a){s - b){s - c)(xviil). In the case of an equilateral triangle this formula becomes^ = Vf Xg- x "g x | = V^ x Q)* = V3x?- = -4336»(xix). 18 MENSURATION. I !< Ill I ii ! 1; Trapezoid. Let b and b' be the parallel sides, then A=^(b + b')p(xx). Quadrilateral. Let d= diagonal, and p and p' the perpendiculars from the diagonal to the opposite angles, then A = ^ (p-p') d (xxi). Quadrilateral in a Circle — i. e., that may be inscribed in a circle. Let a, b, c, d be the four sides, and let 5 = J (a + b + c + d) then we have the formula A = ^ {s-a) (s- b) (s - c) (« - d) (xxii). ]R.eqular Polygon. Letrt = apothem when side is =1, « — a side, and n — number of sides. Then A — \ans (xxiii) ; .'. « = — Cxxi v) ; and n = — (xxv) . ^ an^ «« ^ Circle. Let rf = diameter, r = radius, c = circumference ; and T = 31416. c = itd (xxvi) ; /. d = - =c Xg-j^^ = c X 3183 that \&d=zv^c (xxvii). 4^ \A ^ = Jc = ^(Lm); o'=:oV^ (Liv); «' = «(^)' (lv); - • Z= 2Vo'Tia* (LVi) ; ^ = |afe (lvh). For parabolic zone A = */t( 6' + j pj (lviii), where A = height of zone, and b, 6' = bases or double ordinates. < Htperbola. Symbols same as in ellipse and parabola. = -^(t + a)a (lix) ; a — d±:- where t (Jit ot d = -vie* + o^ (LX) ; c =^ —^ (LXi) ; ea \ c '=7 2 VJc^ + o^ I (LXii), ± according as the smaller or greater absciss is given. 4ca, J. = - |3V7a(7« + 5a) -f- 4V«a| (LXlll). MENSURATION OF SOLIDS, SYNOPSIS OF FORMULAS. Begulab Solids. Let £ = surface, and u== volume or solid contents, and let e = one edge. Tetrahedron or Regular Triangular Pyramid. s = e*V3 = i-732e (lxiv), and v = JgeVS - HTWe (lxv). Hexahedron or Cube, s = qg^ (lxvi) ; v = e^ (lxvii). Octahedron, s = cV^ = 3im^ ( xviii) ; v = ^eV2 ==;-4n406«' (LXIX). ^ ,, . B0UD8. 21 Dodecahedron. » = iVVf(5 4- 2V5) = 2o-.645775e»(Lxx)j V = seViVC^^ + 21V5) = 7-6631c»(LXXl). ICOSAHEDRON. » = 5eV3 = 8-666»(LXXIl) ; V = f e'7(7 + 3V5) = 2.18169e» (LXXIII). Parallelopiped ; Prism ; Cylinder. Let a = area of base or end, p — perimeter of base, and p' = pe- rimeter of section perpendicul?.i to one of the edges of the solid ; also let h = t]xe height, and « = the whole surfuce. v = ah (lxxiv), « = A/) + 2a(LXXv), when the solid i: is right, and s = hp' + 2a (lxxvi), when the solid is oblique. Regular Pyramid and Cone. Let j9 = perimeter of base, i[ = length of slant side, A = height, i. c, perpen- dicular height of vertex above the base, and a= area of base. v = ^ah (lxxvii) ; s = ^l + a (lxxviii). Frustum of Pyramid. Let a and a' ~ areas of the two ends, h — height, e and e' = the edges of the ends, and let p and p' = perimeters of ends. V = I7t(tt + a' + Vaa') (lxxix) , « = i/i ( j ' (LXXX),S^i(j9-|-p')i + a + a'(LXXXl). Frustum op Cone. Symbols as in frustum of pyramid also d and d^ = diameters of ends. V = -^h(a + a, + ^iaaf) (LXXXli). '' Alsoy =-'!8oid(d:' + d;' + dd^^^ = -2QiShid' + d;' + dd^) (Lxxxiii), since a = • '854(i2 and a' = -7854^// and Voa' ^ VC7854ci2 + -7854^^2) g ^ ^,'p + p')l + a + a'(LXXXlv) . 22 MENSURATION Wedoe. Let I and b = length and breadth of back, e ~ length of edge, h = height. Then v = y6A(« + 2/) (lxxxv). Sphere, v - 5236ff»(Lxxx vi), s-ircP- (lxxx vii) . Spherical Segment. Let r= radius of base, rf= diameter of sphere, h = height, and « — convex surface, V = -5236/4 (3,^ + h') (LXXXVIII) ; V = 5236h»(3d - 2h) (LXXXIX), S = irdh (XC). Spherical Zone. v = — (r^ + r/+ l^h^) (xci), where r and r, are the radii of the ends. For middle zone V = "^ (cP + I/O (XCII), V = ~^d,- - -h') (XOIII), where d is the diameter of the end of the zone and d' is the diameter of the sphere, s = irdji (xciv) where s = convex surface. . - * d^h Paraboloid, v = ^ah = ^-^ = 'Sositip/i (xcv). Frustum op Paraboloid. Let a and a' = areas of ends; d and rf, their diameters, and ^ = height, then V = ■sh(a + a') (xcvi) ; v = jirhUP + dj^) = "3927^ (d'' + ^^=')(XCVIl). Spheroid. Let t = transverse, and c = conjugate axes. Then v -- -f^ssGc^^ ^xcviii) for oblate spheroid. v * yr=-523Gc2< (xcix) for prolate spheroid. Circular Segment op Spheroid. jV Oblate V = '5236(30 - 2h) —(c), ' , - Prolate v=.'523G(3<_ 2/t)__(ci) 6 ILLUSTRATIONS AND BXERCISES. 23 EiiLiPTiOAL Segment op Spheroid. ' Oblate V = •5236(3< _ 2/t) _(cii), r Prolate v = -fiaafl (3c - 2/1) — (cm). Middle Frustum op Spheroid. (Circular) oblate v ^ •2018(2<2 + <;»)? (civ), prolate v = :23l8(2c2 + d2)^(cv), (Elliptical w = -2618(2^0 -f-r^)/(CVi) for either oblate or prolate, where Z = length of frustum, (Z = diameter ; and, in (cvi), d and ]{'l — 6) = 90 X V 160 + 90)(lfiO — 90) = 90 X V'.i&<> X 70 =; 90 X Vl»6UU = 90 X 132-28 yards = 11906-2 square yards. Exercise ii. 1. Find the area of a rectangle whose base is 11 and side 16. Ans. 176. 2. Find the area of a rectangle whose base is 28 and diagonal 30. Ans. 301-56. 3. Required the nr.a of a field in the form of a parallelogram whose base is 760 link?, and altitude 250 links. Ans. 1 a. 3 r. 24 per. 4. Required the base of a parallelogram whose area is 2 a. 3 r. 17 per., and perpendicular altitude 120 links. Ans. 2380208 links. 5. Find the diagonal of a rectangle whose area is 200, and base 50. Ans. 50-159. 6. Find the distance between the sides of a parallelogram whose base is 900 yards and area 6 acres 2 r. 28 per. 17 yards. Ans. 35-915 yards. RIGHT-ANGLED TRIANGLE. FoRMULJE. Let h = base, p — perpeiidiculai-, h — hypo- thenuse, p' = perpendicular from right angle on the hypothcnuse, s and «' - the segments into which this 'Ml 26 MEJTSURAMOK. divides the bypothenuse, s being tbat adjacent to tlie base of the triangle ; then h = V^^ + p' (ix) ; l^^h'~p' (X); i,^W~¥ (XI); ,^P" (xn); «= (xui); andj»' = V««' (xrv). Ex. 1. Fifld the bypothenuse of a right-angled triangle whose base is 10 and perpendicular 15, Solution. Here 6 = "lO, and ;> = 15 r then by formula (ix), h =. '^O^^p'i-=z ViOO+!B5 = V = (XVl) ; p = --- p (XV ll) ; A = ^Js{s - a) {s -h (s~c) (xviii), where a, 5, and c are the sides, and s = \(^a -{-1 + c). Aj^o^ for equilateral triangle, J. = 4336^ (xix). Ex. 1. Find the area of a triangle whose base is 91 and altitude 24 chains. Solution. Here 6= 91, and j» = 24; {then by formula (xv), ^ =. i6/> = i x 91 X 24 =1092 obaina =^109-2 acres = 109 acres r. 32 per. Ex. 2. Reauired the area of a triangle whose three sides are 100, 120, aud' 140 links. Solution, Here a = 100, 6 = 120, and c = 140;" thens = J (o + .* + c) = J(100 + 120 + 140 = 180. Then by formula (xviii) A = VlSO x (180 - 100) (180 - 120) ( 180 - 140 =:Vl80x 80x60 X 40= V34560000 = 6878-7 sq. links = -058787 acres =r a. r. 9 sq. per. 12 sq. yards. wm 28 MENSURATION. -hi I ! I ] 1! Tl Ex. 3. Find the area of an equilateral triangle whose base is 1000 yards. Solution. Here 6 = 1000, then by formula (xrx) A — 433&? =-433 X 1000* = -433 x 100000 = 433000 sq. yards = 89 a. 1 r. 34 per. IJ yards. Ex. 4, Find the length of a side of an equilateral garden which contains 4 a. 3 r. .30 per. 19J yards. Solutio7i. Here ^4 = 4 a. 3r. 30 per. lOJ yards = 23917 yards. Then by formula (xix) A = -iHSbi . • b^ = ^, and -433 b — V V A •433 23917 , 433 - V52926 = 230'05 yards. Exercise iv. 1. Find the area of a triangle whose base is 9 and altitude 11. Ans. 49J. 2. What is the perpendicular altitude of a triangle whose base is 750 chains and area 500 acres? Ans. 13J chains. 3. I^nd the area of a triangle whose three sides are 40, 60 and 80 yards. Ans. 1161-8 yards. 4. What is the area of a triangle whose three sides are 420, 480 and 700 links ? Ans. 3 r. 37 per. 24 yaids. 5. The area of an equilateral triangle is 9134 square yards, what is its base? Ans. 145 2 yards. 6. Find the altitude of a triangle whose area is 7196 square feet and base 120 feet. Ans. 1 19 93 feet. 1. 2. TRAPEZOID ; TRAPEZIUM ; QUADRILATERAL INSCRIBED IN A CIRCLE. FORMUL.^. Trapezoid, A = lp{h + 1') (xx) where 6 and t' arc the parallel sides. It: ILLtJSTRAilONS ANl) fikERClSES. e whose base is al garden which trapezium, A = id (p+p') (xxi) ^here p and p' are the perpendiculars from opposite angles to diagonal. Quadrilateral in Circle, A = ^(s - aQs -&)(«- r(« — d) (xxii) Tvhere a, h, c, d, are the four sides, and c = J (a + b + c + d). Ex. 1. What is the area of a trapezoid whose parallel sides are 19 and 25 chains, and the perpendicular distance between them 13 ^chains ? Solution. i Herep = 13, 6 = 19 and »' = 26. Then by formula (xx), Ar=ip{bx 6')= J X 13 + (19 + 26) = i. X 13 X 44 := 286 chains = 28 a. 2 r. 16 per. Ex. 2. What is the area of a trapezium whose diagonal is 700 yards and the perpendiculars from it to the opposite angles, 120 and 80 yards ? Solution. I Here d = 700, jj=<120 and p) = 80, then by formula (xxi), A = id{p-^p) = i X 700 X (120 4- 80) = } X 700 X 200 = 70000 square yards = 14a, 1 r, 86 per. 1 yard. Ana. Ex. 3. Find the area of a field in the form of a quadrilateral [whose opposite angles are equal to two right angles ; i. = i X 38 X 777 = 1476-3 squara feet .-. A of smaller segment — 17872 — 1476-3 =: 3109 square feet. Hence area of lune = A~A'z= 1352-827 — 310-9 = 1041-927 square feet. Exercise x. 1. What is the area of a sector whose arc contains 36° and who8» rildius is 3 feet? Ans. 2-8272, jih i:i 8e MENSURATION. 2. What is the area of a circular sector whose arc is 6fi0 feet in length and whoso radius is 325 feet ? Ans. 106625 sq. feet* 3. Find the area of a segment of a circle, the arc coQtaiiiiDg 280°| the radius being 5 feet and apothem 3 feet. Ans. 73 082. ELLIPSE. FoRBlULiG. Lot C= circumference, < = transverse axis; c = conjugate axis, a = absciss, o = ordinate. Then C= » y • -^ (xlvii) A=-~= -isutc (xlviii) ; t = — V(* - «)« (XLix) ; a = "5 ± c2 and d' ot C~ ,-- === (LIl). Ex. 1. Find the transverse axis of an ellipse whose conjugate axis is 15, an ordinate 6 and the smaller absciss 9. Solution. ca (c fc^ ) 15 X 9 f 16 . /225 „ ) = WP^ + VV} = -^4' C/ + t) = \^ X 12 = 45. Ex. 2. Find the ordinate of an ellipse whose axes are 45 and 15 and one absciss 9. ^ Solution. By formula (xlix), o=z-^(t — a)a = i|- X <^(^—9) x 9 = J y86 x ? = i X yaa4 ^ ^- x is = §. ^ . ILLUSTRATIONS AND EXERCISES. 87 Ex. 3, Find the area of an ellipse whose axes are 30 and 40. Solution. By formula (zLviii(, ^« — =-7864x80x40=d42-48. Exercise xi. 1. Find the circumference of an ellipse whose axes are 20 and 16. Ans. 56-8943. 2. What are the abscisses of an ellipse whose axes are 80 and 120 and an ordinate 25 ? Ans. lOG-836 and 13-163. 3. What is the urea of an ellipse whose axes are 28 and 20 chains ? Ans. 43 a. 3 r. 37 per. 5 yds. 4. What is the ordinate of an ellipse of which the axes are 25| and 18 J and one absciss 7^? Ans. 8-429. 0. What is the area of an elliptical park of which the conjugate axis is 1800 links, an ordinate 400 links, and the smaller absciss 600 links ? Ans. 162 a. 3 r. 10 per. 28 yds. 6. What is the area of an ellipse whose transverse axis is 100, an ordinate being 20 and the greater absciss 7o? Ans. 3627-44. PARABOLA. Formulae. Let p = parameter, a and a' = ahy two abs- cisses, and o' their corresponding ordinates, b = base or double ordinate, and Z = length of parabolic curve. Theni> = ^(LlTl);o' = oV (~j(LiY)]a' = a(^J (LV) ; Z = 2V(o» + |a») (LVi) ) A = ^ab (lvii). For parabolic zone, A = ^hlb' + ~ — ~, ) (lviii) where h, = height of zone apd b and 6' = bases or double ordbfttes. I I 88 MENSURiLTION. Ex. 1. Find the parameter of a parabola whose ordinate is 25 and absciss 12. Solution. o» 25'* 626 , Bjr formula (Liii),p«— = » — ™5anr. a 12 11 Ex. 2. Find the areu of a parabola whose base or double ordinate is 30 and height 22. Solution. By formula (Lvm), ^— ^«i6=ax80x22«.440. Ex. 3. Find the length of a parabolic curve of which the ordinate and absciss are respectively 30 and 8. Solution, By topmula v^-vi), I - 2^oS + Ja* = 2^900 + ^ x 84 • 2^900 + l^fi -= 2yiTa6 = 4^8868 = § X 94-17 •= 6278. EXEHCISE XII. <» Jl 1. Given an ordinate of a parabola, 60 and its absciss 42, find, the parameter. Ans. 85-7. 2. Two ordinates are 40 and 30 and the absciss of the former 21, find that of the latter. . Ans. 11 -8125. 3. Find the area of a. parabola wLose base is 75 and height 48 chains. Ans. 240 acres. 4. Find the area of parabolic zone whose parallel ends are 12 and 16 and height 8. Ans. 112-76. 6. Find the length of a parabolic curve whose absciss is 12 and ordinate 15. Ans. 40-841. 6. What is the ordinate of a parabola whose absciss is 20 j a second absciss and ordinate being 6 and 4 respectively. Ans. 7-302. ate is 25 ordlnat* hich th« ^800 + tffi !3 42, find, Ans. 85-7. former 21, 18. 11-8125. height 48 . 240 acres, are 12 and Ln3. 112-76. I is 12 and ^ns. 40-841. iss is 20 : a 'I- ILLUSTRATIONS AND BXBRCISBS. HYPERBOLA. ^ Symbols same as for ellipse and parabola,. Formula o = ~-^(t + a)a (lix); a-d ± 3a t and J t I f.1 of d=-y -r + o» (Lx) ; c = ; .7=7~T" '^ 4 V -^ ' V (^ + a)a ca { c ./ J ) 2 (LXI) ; »c«. ^ = 75#^^«(^^ + ^«) + 4V/"a I (LXIII) . 1. What is the ordinate of an hypprbola of which the axes ar« 30 and 15, and the smaller absciss 10 ? v« JSolution. By formula (lix). o = ^- V(t +~r7~fiT7~x 60-Hx'7'5) + 4V60l-lW-9». 40 MENSURATION. Exercise xiii. 1. Find the transverse axis of an hyperbola whoge ordinate is 20, smaller absciss 10§, and conjui^ate axis 30. Ans. 50. 2. What are the abscisses of an hyperbola whose axes are 30 and 25 and the ordinate 16? , Ans. 39-36 and 9-36 3. What is the area of an hyperbola whose axes are 45 and 90, the smaller absciss being 30 ? 4. The conjugate axis of an hyperbola is 45, the ordinate 90, and the smaller absciss 7^, find the transverse sixis. Ans. 22-5. . 5. The axes of an hyperbola are 15 and 20, and an ordinate 10, find the abscisses. Ans. 26^ and 6|, 6. Find the area of an hyperbola whose axes are 55 and 33 chains, and smaller absciss 18J chains. Ans. 50 a. 3. r. 37 per. W'' m I r MENSURATION OF SOLIDS. 2%e Five Regular Solids. Lefc s = surface, v = volume or solid contents, and e = one of the edges. Tetrahedron or Regular Triangular Pyramid. «=cV3= l-732e (lxiv) ; and i;=ieV2=-11785fl3 (lxv). Hexahedron or Cube. s~Qe* (lxvi); v = e^ (lxvii). Octahedron. a=26'«V3 — 3-464c« (lxviii) ; i;=Je3V2 = •471405c3 (txix). Dodecahedron. a=15e Vi (8+2V5> = 20-645775e« (Lxx) ; ' ' 40 ' i ..-, m SOLll>d« 41 37 per. loOSAHEDRON. g = 6««V3 = 8"66e« (tJtxil) ; V - f c' Vi^T+lys) = 2-18l69e' (Lxxiii). Ex. 1. Find the surface and volume of a tetrahedron whose edge is 20 feet. Solution. By formula (lxiv), s = l-732c8 = 1-732 X 20* = 692-8 square feet. By formulj., (Lxv), v = -11785e3 = -11786 X 203 = 942-8 cubic feet. Ex. 2. Find the surface and solidity of a hexahedron or cube whose edge is 9 feet. Solution. By formulas (lxvi) and (r,xvii), s = Gc* = 6 x 9* ^486 square feet, and « = e3 = 93 = 729 cubic feet. Ex. 3. Find the surface and cubic contents of a dodecahedron whose edge is 4 feet 2 inches. Solution. By formula ri-xx;, s= 20-645775e2= 20-645785 X 502 (-81006 4 ft. 5 in. = 60in.)= 51614-4876 square inches = 368-488 square feet. By formula fLxxi), v = 6-6631e3 = 7-6681x503 = 957913 76 cubic inches = 564-849 cubic feet. Exercise xiv. 1. Find the surface and cubic contents of an icosahedron whose edge is 4. Ans. 138-56 and 139-628. 2. Find the surface and [solidity of a cube or hexahedron whose edge is 20. Ans. 2400 and 8000. 3. Find the surface and volume of a tetrahedron whose edge is 8. • Ans. 110-848 and 60-339. 4. Find the surface and solid contents of a dodecahedron whose edge is 10. Ans. 2064-5775 and 76631. 5. Find the surface and volume of an octahedron whose edge is 1 1. Ans. 419-144 and 627-44. 6. Find the surface and cubic contents of an icosahedron whose edge is 5 yds. Ans. 216-506 sq. yds. and 272-711 cub. yds. 553S5SSS7SFm 42 MENSURATIOir. BIOHT AND OBLIQUE PARALLELOPIPEDS, PRISMS, * CYLINDERS. Let a = area of base or end, p = perimeter of base, and p' = the perimeter of a section perpendicular to one of the edges of the solid j also let h = the height, s = the whole surface. Then v = ah (Lxxiv), s — hp + 2a {lxxy) when the solid is right, s-hp' + 2a (LXXVi) when the solid is oblique. Ex. 1. What are the surface and volume of a prism whose height is 20 feet and base an equilateral triangle, each side of which is 24 leet? Solution. By formula fxix;, a = -48362 =r -433 x 4« = 6-928 square feet = area of base and perimeter = 4 x 3 = 12. Theu by formula (lxxy), « = *p + 2a = 20 x 12 + 2 x 6-928 =240 J- 13-856 = 263-85H square feet — Burl'uce. Also by formula f lxxiv;, t> »^ aA = 6-928 x 20 = 138-56 cubic feet. Ex. 2. Find the surface and solid contents of an oblique prism whuse base is a regular hexagon, with edge = 10 inches, the lateral edges of the prism being 40 feet long and the perimeter of a section perpendicular to them 4^ feet. 1. 3. i. I:! Solution. By formula f xxin;, area of base = A — iana = i x 8-66026 x 6 X 10 ac 269-8075 square inches = 1'8(.42 square feet. Then • = Af>' + 2a = 40 X 4i + 2 X 1-8U42 = 180 + 3-6084 = 188-6084 sq. ft. Also v = aft = 1-8042 X 40 = 72*168 cubic feet. Ex. 3. Find the surface and solidity of a riglit cylinder whose height is 20 feet and diameter 12 feet. Solution. Byformnla (xxxi;, area of the ba«o=^= Tr«=81416x««=ll«"0976 sq. ft. ▲lM> c s 12 X 8-1416 = 87 6092 fast. ih >.-. v.- SOLiDfi. 4S ■then by formula (lxxt), » =ftp J- 2a = 20 x 87-8992 + 2 x 118*0976 a= 763 984 + 228- 1952 = 9831732 sq uare feet. Also r — oA = 1130976 x 20 = 226l-95'4 cubic feet. Bz. 4. How many gaHons of water will a cylindrical cistern contain, whose diameter is 6 feet and depth 7 feet? Solution. Area of base = 31416 X 32 = 37'2744 square feet. Hence volume —ahz= 372744 < 7 = 260.9208 cubic feet. Then since each cubic foot contains 6^ gallons x6i' — 162d'88 gallons. Exercise xv. 1. Find the surface and cubic contents of a rectangular parallele- piped whose height is 25 feet, its base being 4 feet wide and 5 feet lorg. Ans. 490 sq. ft. ; 500 cub. ft. i. How many gallons of water are contained in a circular cistern whose diameter is 12 feet and depth 10 feet ? Ans. 7068-6 gallons. 3. Find surface and solidity of a right prism whose base is an octagon, each side of which is 2 feet, the edges of the prism being each 18 feet long. Ans. 326-6274 sq. ft. and 3476448 cub. ft. 4. Find the surface and solidity of an oblique prism, each end being a regular nonagon whose side is 20 inches, the edges of the prism being 14 feet long and the perimeter of a sec- tion perpendicular to them being 13 feet. Ans. 216 3434 sq. ft. ; 2404038 cub. ft. 9. How many pails of water are contained in a pentagonal cistern whose depth is 15 feet, each edge of the bottom being 6 feet ? , Ans. 2322-6446 pails. Note. A pail holds 10 quarts. f. Find the surface and solidity of a right cylinder whose height is 42 fti«t and circumference 22 inches. Ana. 77-535 sq. ft. and 11*2368 cub. ft. i 44 MUNSURATlON. J »' = peri- meters of ends; and, iu case of cone, d and d' 'Jaa ' = diamoters of ends. i;-J/i(a + a' + V"-"') (lxxix) ; , , fae - a'e'\ , ^ . v^'-ZGiShicP + d'^ + dd,) (LXXXlll); - s — ^(p+p')l + a + a' (lxxxi). Ex. 1. Find the surface and solidity of aright cone, whose slant side is 20 feet, the diameters of the end being 4 and 2 feet. Solution. Here the radii are 1 and 2 feet and thpir difTerpncp is 1 foot. Hence heigiit = -^20-2"— r(r=V400 — 1 =r V39J^19 9749. By Formula (r.xxxtv), v = •261S/i(rf>' +(1/^ + dd,) = •2618 X 19 9749 X (2a + 4i + 2 X 4) = -2618 X 19-9749 X 28 = 146-424 cub. feet. By formula (Lxxxi) s — Up -^ p')l + a+if/ = i X 18-849 5 X 20 + 12-50C4 +31416 = 188-496+15-708= 204-204 sq. ft Ex. 2. Required the surface and solidity of a frustum of a regu- lar hexagonal pyramid, the sides of its ends being 6 and 4 feet res- pectively, and its length 24 feet. Solution. By formula (xxiii), areas of ends = ^ana = ^ X 6-19615 X 6 X 6=93'6307; and i X 3-4641 X 6 X 4 = 41 5692. Difference of apothems of hexagons whose sides are 6 and 4 feet=:6'1961 and 3-4641 = 1-732. Hence by formula (xi), h(>li»ht of frustum = V2i2 — VTd'li' = V576— T(nearly ) = VoTs = 23937. Then by formula (lxxix), v ~ J x fe X (a + o, + ^~. = J X 23-937 X (93 5307 + 41-6392 + ^93 5307~xn-5692r - 7 979 X (135-0^99 = V 38"8?«98374r=. 7.979 + 135-09C9 + 62-368) = 7-979 X 197-4629 = 1576-476 cub. ft. By formula (Lxxxr), s ~ ^(83 -f 24) x 24 + 93-5307 + 41-5692. = 30 X 24 + 136-0999 =. 720 -f 135-0999 =856-0999 sq. ft. 46 MENSURATION. I WEDGE. FoRMULiE. Let I ~ length of back and fe = breadth of * back, e = length of edge, and A = height. Then v = \ lh{e + 21) (Lxxxv). Ex. 1. The length and breadth of the base of a wedge are 70 inches and 15 Inches, the edge is 110 inches in length, and the height is 17-145 inches; what are its solid contents ? Solution, Bereft ■ 15 in = 1-25 ft. ; A = 17145 in = 1-42876 ft. ; « = 110 in « 9^ ft. ti Ifl if f ;; i and ^ = 70 in = 6^ ft. By formula (lxxxv). t;=^6A{e + 2/(= ^ x 1-26 Xl-42876) X9^ +5g X 2) = ^ X 1-26 X 1-42876 X 20| = 6-767 cub. ft. . Ex. 2. Find tlie solidity of a wedge whose base is 6 inches long and 4 wide, its edge being 16 inches in length and height 15-8745 inches. Solution. By formula (lxxxv), v = ^bh(e + 2^ = ^ x 4 x 16-8745 X (16 + 2 X 6 = ^ X 4 X 16-8746 X 28 =ix 168746 X 66 = 66 X 52916 = 286-324 cub. in Exercise xvii. 1. Find the solid contents of a wedge whose length is 64 inches, the edge being 42 inches long, the base 9 inches broad, and the height of the wedge 28 inches. Ans. 4 cub. ft 288 cub. in- 2. Find the solid contents of a wedge whose height is 20 inches, the base 12 inches wide and 10 inches long, the edge being 24 inches. Ans. 1 cub. ft. 432 cub. in. 3. Find the solidity of a wedge whose edge is 2-7 feet long, and back 3-2 feet long, the breadth of the back being40 inches and the height of the wedge 4 feet. Ans. 20 cub. ft 384 cub. i». SOLIDS. SPHERE, SPHERICAL SEGMENT. Formula. Let rf= diameter. Then For sphere; i; = -5236d» (lxxxvi) ; and « = »

*2 + h'^) (lXXXVIII) ; v= 6226^2 (3 iJ MEirStJRATIOy. paraboloid; frustum op paraboloid. Let a = area of base of paraboloid and h = height ; also let a and a' = areas of ends of frustum, and d and dt their diameters. ird'h Then for Paraboloid, v = ^ah= -q- = saizi = -5236 X 39 x25 = 610-51 cub. in. 626X9 — = •5236(3 X15-2X3)X C2 226 52 MENSURATION. Ex. 3. Flna me solidity of an elliptical segment of a prolate spheroid whose height is 10, the axes being 100 and 60. Solution. "^' 100x102 Bj forAuIa foin), v = -5233(30 — 2/t) — =r -5135(3 X 60 — 2 10)x = -6236 X 160 X J^Ji fl = 139323. eo I m til i „. I', I ! j I'l*! ^1: Exercise xxi. 1. Find the solid contents of a prolate spheroid whose diam'^tera are 12 and 16 feet. . Ans. 1206-3744 cub. ft. 3. Find the solid contents of a circular segment of a prolate sphe- roid whose diameters are 24 and 40 inches, the height of the seofuaent being 4 inches. Ans. 3377748. 3. Find the solidity of an elliptical segment of an oblate spheroid whose diameters are 20 and 24, the height of the segment being 5 inches. Ans. 2028-95 cub. in. 4. Find the solidity of an oblate spheroid whose diameters are 16 and 2u inches, Ans. 5663 2576 cub. in. 5. Wbat is the volume of a circular segment of an oblate spheroid whose diameters are 10 and 16 inches, the height of the seg- ment being 4 inclies. Ans. 4718264 cub. in. What is the volume of an elliptical segment of a prolate sphe- roid whose diameters are 11 and 15 feet, the hcij;ht of the segment being 6 feet ? Ans. 539-884 cub. ft- 6 MIDDLE FRUSTUM OF SPHEROID. Let t = length of frustum, and d — end diameter. Then for circular frustum of oblate spheroid, V = -2618/(2^ + cP) (civ) ; For prolate spheroif? V =^ -26i8?(2c^ + (P) (cv.) For elliptical frustum let d and d^ = diameters of ends, then whether the frustum is a portion of an oblate or prolate spheroid, # = -^aislijc + dd,) (cvi.) SOLIDS. 53 Ex. I. Find the solid contents of the middle circular frustum of an oblate spheroid, the axis of the spheroid being 25 inches, the end diameters 20 and the length 9 inches. Solution, Bj formula (civ), v = -2618/(2/2 ^. rf?) = -2618(2 x 95« + 20«) x 9 = -2618(1250 + 400) 4- 9 = -2618 X 1660 X 9 =3887-78cub. in. Ex. 2. Find the solid contents of an elliptic middle frustum of a prolate spheroid whose axes are 24 and 30, the end diameters being 16 and 20, and the length 10 inches. Solution. By formula (cvi>, v = -2618(2/0 + ffd')f = -2618(2 X 30 X 24 + 16 X 20) )< 10 = -2618(1440 + 320) X 10 — -2618 X 1760 X 10 = 4807-68 cub. In. Exercise xxii. 1. Find the solid contents of a circular middle frustum of an ob- late spheroid whose middle axis (i. e. the transverse axis) is 20, the diameter of the end being 14 and the height 10. Ans. 2607-528. 2. Find the solid contents of an elliptical middle frustum of a spheroid whose axes are 30 and 50 inches, the end diameters of the frustum being 18 and 30 inches and its length 40 inches. Ans. 21 cub. ft. 782-88 cub. in. 3. Find the cubic contents of a circular middle frustum of a prolate spheroid whose middle or conjugate axis is 20 inches, the end ' diameter of the frustum being 15 and its length 30 inches. Ans. 8050-35 cub. in. hyperboloid; frustum of htperboloid. Let r = radius of base, and — the diameter of the circle being 60 feet. Ans. 314- 16 sq. feet. 24. Find the area of a circular annulus-rthe circumferences of the containing circles baing 90 and 60. Ans. 358-2. 25. The diameters of two concentric circles are 50 and 30 feet — find the area of the included annulus. Ans. 1256 64. 26. What is the area of a triangle whose basis is 12^ chains and altitude 8f chains ? Ans. 5 a. r. 33 per. 27. What is the area of a trapezoid whose parallel sides are 7J chains and 12^ chains, the perpendicnlnr distance between them being 15^ chains. Ans. 15 a. r. 33 per. 6 yards. 28. The circumference of a circular fish pond is 400 yards — what is the side of a square pond of equal area? Ans. 112-85 yards. 29. What is the area of a triangle whose sides are 24, 36 and 48 yards respectively? Ans. 418282 sq. yards. 30. Find the area of a square field whose side is 19 chains. Ans. 36 a. r. 16 per* 31. Find the area of a triangular field whose three sides are respec- tively 120, 140, and 160 yards. Ans. 1 a. 2 r. 28 per. 26 yards. 32. Required the area of a field in the form of a rectangle whose adjacent sides are 740 yards and ISO yards. Ans. 27 a. 2 r. 3 per, 9 yards. 33. What is the area of a circle whose circumference is 92 ? Ans. 673-734. 34. Find the area of a quadrilateral inscribed in a circle — its four sides b«ing 400, 360, 300, and 280 links. Ans. 1 a. 15 psr. 26 yards. MISCELLANEOUS EXERCISES. 67 35. Find the area of an equilateral triangle whose base is 20. Ans. 173-2. 36. Find the area of a circle whose radius is 35. Ans. 3848-46. 37. Find the area of a quadrilateral whose diagonal is 81 chains and perpendiculars from it to the opposite angles 29 chains and 23 chains respectively. Ans. 208 acres. 38. Find the area of a trapezoid whose parallel sides are 750 and 600 links, and the perpendicular distance between them 240 links. Ans. 1 a. 2 r. 19 per. 6 yards. 39. Find the area of a triangle whose sides are respectively 90, 70, and 60 chains in length. .Ans. 209 a. 3 r. 1 per. 6 yds. 40. A circular garden is to be formed so as to contain as much land as an equilateral triangle who3e side is 56 chains. Re- quired the diameter of the circular garden and also its ares. Ans. 914-76 ; 135 a. 3 r. 6 per. 6 yds. 41. Find the area of a circular annulus contained between two circles whose diameters are respectively 100 and 160. Ans. 12252-24. Required the length of a circular arc of 68°, the diameter of the circle being 250 feet. Ans. 148 34. Find the area of the sector of a circle whose radius is 50 feet, the arc of the sector containing 70". Ans. 152705. Find the area of the segment of a circle whose diameter is 60 chains, the circular arc containing 130", and its chord being 52 chains in length. Ans. 63 a. r. 29 per. 6 J yds. Find the area of a regular decagon whose side is 11 and apothera 9-526279. Ans. 523 9453. 46. Find the area of the sector of a circle whose radius is eoyards, the arc of the sector being 280 yards in length. Ans. 1 a. 2 r. 37 per. 20 yds. 6 ft. 47. Find the area of a field whose opposite sides are parallel, the base beimg 620 yards, and the perpendicular altitude 108 yards. Ans. 13 a. 3 r. 13 per. 16 yards. 48. What is the length of an aic of 197} ® of a circle whose dia- meter is 240 yards ? Ans. 413 6124 yards. 48. Required the circumference of an ellipgd whose diameters ara 60O and 400. -^QS- 1570-8. 42. 43. 44. 45. 68 MENSURATION. \ I I |1" r m 3 I ■i ill: [l 111 50. Afield containing 7 a. 3 r. 21 per. 17 yds. is divided into two parts, the one forming a circle whose diameter is 80 yards, what must be the dimensions of an equilateral triangle whose area shall be equal to the remainder ? Ans. 276-63 yards. 51. Find the area of a circular annulus contained between two circles whose circumferences are 360 and 240 chains. Ans. 573 a. r. 19 per. 6 yards. 52. What is the area of an ellipse whose diameters are 5 and 10 ? Ans. 39-27. 53. The axes of an ellipse are 30 and 10, and one absciss is 24 ; what is the ordinate? Ans. 4. 54. The axes of an ellipse are 70 and 50, and an ordinate 20 ; what are the abscisses ? Ans. 56 and 14. 55. The conjugate axis of an ellipse is 10, the smaller absciss 6, and the ordinate 4 ; what is the transverse axis? Ans. 30. 56. The transverse axis is 280, and ordinate 80, and one absciss 56 ; what iti the conjugate axis? Ans. 200. 57. If an ordinate of a parabola is 20 and its absciss 36, what is the parameter? Ans. 11*1. 58. The two abscisses are 9 and 16, and the ordinate of the former is 6 ; find that of the latter. Ans. 8. 59. Given the two ordinates 6 and 8, and the absciss of the former 9, to find that of the latter. Ans. 16. 60. Find the area of a parabola whose base or double ordinate is 15, and height or absciss 22. Ans. 220. 61. Required the length of a parabolic curve whose absciss is 6 and ordinate 12. Ans. 27-71. 62. The transverse axis of an hyperbola is 15, the conjugate axis 9, the smaller absciss 5, required the ordinate. Ans. 6. 63. The transverse and conjugate axes of an hyperbola are 60 and 45, and one ordinate is 30 ; what are the abscisses ? Ans. 67^ ahd 7^. 64. The transverse axis of an hyperbola is 60, an ordinate 24, and the smaller absciss 20 ; what is the conjugate axis ? Ans. 36. i' MISOBLLANEOUS EXERCISES. 69 65. Tbo conjugate axis of an hyperbola is 45, the smaller absciss ' 30, and the ordinate 30 ; what is the transverse axis ? Ans. 90. 66. What is the area of an hyperbola whose axes are 15 and 9, and the smaller absciss 52? Ans. 37-9i9. 67. Find the volume and surface of a tetrahedron whose edge is 8. Ans. 60-3 and 11085. 68. Find the volume and surface of a hexahedron whose edge is 11. Ans. 1331 and 726, 69. Find the volume and surface of an octahedron whose edge is 10. Ans. 471-4 and 346-4. 70. Find the volume and surface of a dodecahedron whose edge is 4. Ans. 490-44 and 330*33. 71. Find the volume and surface of an icosahedron whose edge is 6. Ans. 471-245 and 311-76. 72. What is the surface of a right cylinder whose length is 20 and circumference 6 ? Ans. 125-73. 73. What is the surface of a regular pentagonal pyramid, each side of its base being 1§ feet and its slant side 10 feet in length? Ans. 46-4456. 74. Find the surface of a frustum of a right cone, its length being 31, and the circumference of its two ends being 62832 and 37-6992. Ans. 198549. 75. What is the surface of a sphere whose diameter is 800 inches. Ans. 2010624 sq. inches. 76. Find the surface of a globe whose diameter is 13 and circum- ference 37-6992. Ans. 45239. 77. Find the surface of a spherical segment whose height is 2, the diameter of the sphere being 10. Ans. 62-832. 78. What is the volume of a prism whose length is 18 feet, its base being a regular hexagon whose side is 16 inches and apothem 13-8564 inches ? Ans. 83-138 cub. feet. 79. If the volume of a triangular prism is 7-656 and its length is 10} ; what is the area of its base ? Ans. 729. 80. Required the volume of a frustum of a square pyramid, the side of the^ greater base being 16, of the lesser 10, and its length 18. Ans. 37-152. 60 MENSURATION. *' Hi [ I is < w r 81. What is the solidity of a cone whose altitude is 12 feet, th« diameter of its ba^e beiog 10 fuet? Ans. 314-lt>. 82. Find the area of the base of a cone whose volume is 282-74 and altitude 30. Ana. 28274. 83. What is the solidity of a sphere whose diameter is 30? Ans. 14137-2, 84. What is the diameter of a sphere whose volume is equal to 65449-85 feet? Ans. 50 feet. 85. What is the solidity of a segment of a sphere, the height of the segmeat being 2, the diameter of the sphere 10 ? Ans. 54-4544. 86. What is the volume of a spherical segment, whose height is 10, and the diameter of its base 20? Ans. 20944. 87. Find the volume of a spherical zone, the diameter of its end being 10 and 12, and its height 2. Ans. 195-9159. 88. Required the solidity of the middle zone of a sphere, its height being 32 feet, and the diameter of the sphere 40. Ans. 31633-8. 89. Find the volume of the middle zone of a sphere, its height being 8, and end diameters 6. Ans. 494278. 90. Find the solidity of an oblate spheroid whose axes are 20 and 12. Ans. 2513 28. 91. What is the volume of a prolate spheroid, its polar axis being 7, and equitorial axis 5 ? Ans. 91-63. 92. Find the area of the segment of a circle whose radius is 40 yards — the arc containing 136°, the chord being 60 yards in length. Ans. 11050523 yds. 93. What is the transverse axis of an ellipse whose conjugate axis is 90 and area is equal to that of an equilateral triangle, whose side is 70 and a circle whose circumference is 240 ? • Ans. 94-87. 94. Find the area of an equilateral triangle whose side is 90. Ans. 35 W -3. 95. What is the area of a triangle whose ^ides are 48, 54, and 60 respectively? Ans. 1231-09.. 96. Find th« area of an ellipsa whos« diameters are 40 and 48. Ans. 1507-968. 9T 98 89 10 10 10 10 u w Ifl 1( ^.r MISCELLANEOUS EXERCISES. 61 97. Find the length of a rectangular field whose breadth is 220 yards, and which contains as ranch ground as an ellipse whose axes are 900 and 1100 yards. Ana. 3534-3 yards. 98. Find the diameter of a circle whose area is 5 acres, 3 roods, 27 per., 20 yards. Ans. 191 04, 99. Find the altitude of a parallelogram whose base is 500 yards, and area equal to the combined areas of a circle whose cir« cumference is 200 yards, and a circular sector whose art contains 200^ and whose radius is 40 yards. Aus. 11-95 yardf. 100. Find the area of a sector, a circle whose radius is 300 link^ the arc being 600 links in length. Ans. 3 roods. XOl, Find the solidity and surface of a hexahedron whose edgt is 7. Ans. 343 and 294. 102. Find the lurfaee and relume of a dodecahedron whose edgt . is 4. Ans. 330-3324 and 490 -4384. lOS. Find the surface and solidity of a cone whose height is 20 feet, and diameter of base 10 feet. Ans. 402-36 and 523-^ lOi, Required the aurface and solidity of a right prism whose bast is a regular heptagon, baring each of its sides 8 feet, ifaH edges of the priam beiag eaeh 8| feet in length. Ana. M114 and 813-9963 Mn!^ How many pails of water (each containing 10 qts.) may bB contained in a circular cistern whose diameter is 7 feet, and depth 11 feet? Ans. 1058-3265 pails. VD/S, What must be the depth of a pentagonal cistern which coi^ tains as much water as a circular cistern 8 ft. in diameter and 4^ feet deep, and a rectangular tank 7 feet long, 5 feet wide and 3^ feet deep — one side of the pentagonal cistem being 5 feet. Ans. 12-667 feet 107, Find the surface and solidity of a pyramid whose height is 9^ feet — the base being a regular hexagon whose edge is 3 feet Ans. 10769 and 70- 148. 108. Find the surface and solidity of an oblique prism whose ham is a pentagon with each edge 4 feet— the edges of the prism being each 10 feet long, and the perimeter of a section pe»> pendicular or to them 18 feet Ans. 235065 sq. ft. 275-276 eub. £1;. B 62 MENSURATION. 1^ 109. Find the area of triangular field nrhose sides are 8, 12 and 14 chains. Ans. 4 a. 3 r. 6 per. 15 yds. 110. Find the surface and solidity of an icosahedron whose edge ii ■A feet. Ans. 77-94 and 58-90563. 111. Find the solidity and surface of the frustum of a right cone whose slant side is 60 feet — the diameters of the ends being 10 and 2U feet. Ans. 10957 11 cub. feet and 3220- 14 sq. ft 112. Find the solidity and surface of the frustum of a right pyramid whose ends are squares with edges 10 and 12 feet respectively and height 8 feet. Ans. 970-66 and 508-653. .113. How many cubic feet are there in a squared stick of timber whose end edges are respectively 28 and 20 inches, th length of the stick being 42 feet ? Ans. 169 cub. feet. ill4. What is the solidity and surface of a hexagonal frustum whose height is 6 feet, the edges of the enas being respec- tively 2 feet and 1^ feet? Ans. 48*064 cub. feet; 63- 147 sq. feet. U15. Find the area of a triangular park whose three sides are 900, 1100 and 1300 links respectively. Ans. 4 a. • r. 20 per. 27 yds. k116. Find the diameter of a circle which shall contain as much ground as a quadrilateral inscribed in a circle, whose four sides are 900, 1000, 60U and 800 yards respectively. Ans. 916-48 yardf. 117. Find the area of a square whose diagonal is 44. Ans. 968 sq. yds. dl8. Find the area of an annulus inclosed between two concentric circles whose circumferences are 180 and 225 yards respec- tively. Ans. 1450-71 sq. yds. .119. Find the area of an elliptical field whose diameters ate 980 and 1250 links. Ans. 9 a. 2 r. 19 per. 11-6 yds. .120. Find the area of a parabolic zone whose height is 25 yards, its double ordinates being respectively 90 and 70 yards. Ans. 2010-416 sq. yds. .121. Find the surface and solidity of an icosahedron whose edge is 8^ feet. Ans. 625-685 sq. ft. ; 1339-83 cub. ft. ^^^> MISCELLANEOUS EXERCISES. 68 122. Find the solidity of an oblique triangular prism, the edges of the ends being 10, 16, and 24 feet, the height 20 feet. •^ • Ans. 1161-894 cub. feet. 123. Find the surface and solidity of a right cone whose height is ! 20 feet— the diameter of the end being 12 feet. Ans. 606-68 sq. ft. ; 753*984 cub. ft. 124. Find the solidity of a prolate spheroid, whose axes are 11 and 7 respectively. Ans. 282-22. 126. Find the solidity of an oblate spheroid, whose axes are 20 and ■'^'.:' 16 respectively. ' Ans. 3141.6. 126. Find the surface and solidity of a sphere, whose diameter is 26 feet. Ans. 2123-7216 sq. ft. ; 9202-7936 cub. ft. 127. Find the surface and solidity of a spherical segment, whose height is 2 inches, the diameter of the sphere being 6 inches. Ans. 31-416 sq^-in. ; 230384 cub. in. 128. Find the solidity of the middle zone of a sphere, the diameters of its ends being 7 feet and its height 6^ feet. Ans. 393-943 cub. ft. 129. Find the convex surface of a spherical segment, vrhose height is 9 inches, the diameter of the sphere being 3 feet 6 inches. Ans. 1187-52 sq. in. 130. Find the surface and solidity of a frustum of a right cone whose height is 9 feet, the diameters of the ends being 10 feet and 6 feet. Ans. 461-815 cub. ft. ; 461-81 sq. ft. 131. Find -the solidity and surface of an octagonal pyramid whose height is 8 feet, each edge of the base being 5 feet. Ans. 321-89 cub. ft.; 32113 sq. ft. 132. Find the area of a field in the form of a circle, having a diameter of 11 chains. 64 links. Ans. 10 a. 2 r. 26 per. 19-23 yds. 133. Find the area of a triangle whose three sides are respectively . 79, 80, and 90 yards long. Ans. 2 r. 8 per. 21 yds. 1-8 ft. 134. Required the area of a quadrilateral field whose diagonal is 29 chains, the perpendiculars upon it from the opposite " ■ angles being 9 and 17 chains respectively. 'A'--^ , ■ Ans. 37 ft. 2 r. 32 per. 64 MENSURATION. lU 'T m HI 1 1 116. What is tht'area of ft regular nonagon whose side Is 13 yards? Ang. 1 r. 37 per. 2 yds. 1-6 ft 116. Find the solidity and surface of a right cone whose height It 12 feet, the circumference of the base being 31-416 feet. Ans. 31416 cub. ft. ; 382-74 sq. it 137. Find the solidity and surface of a hexagonal pyramid whoM height is 24 feet, each edge of the base being 7 feet. Ans. 1018 44 cub. ft ; 64713 sq. ft 1|8. What must be the diameter of a circular garden to contain m much ground as a field in the form of an equilateral triang^a^ whose side is 250 yards long 7 Ans. 185*6 ydi. |9. Find the are* of an annulus contained between two eono trio circles, whose diameters are 12 and 16 feet. Ans. 63-6174 sq. ft Ujp. Find the areft of a circular sector whose arc contains 40 de» grees, the diameter of the circle being 20 yards. / ns. 34-906 sq. ydt. 1^1. Find the rolume of a spherical sone, the diameters of its ends being 20 and 28 inches, and its height 7} mches. Ans. 37080697 cub. 1b. 1|2. Find the area of a sector whose are is 500 links long, th« diameter of the circle being 500 links. Ans. 2 roods, 20 pen 113. Required the solidity of a cone whose height is 12 feet, tba circumference of the base being 50 feet Ans. 796 cub. ft t^A. Required the solidity of an oblique octa'»onal prism, each side of the base being 9 feet, the height of the prism being 12 feet and each edge 20 feet, the perimeter of a section per- pendicular to the edges being 60 feet Ans. 6008-7 cub. ft Find the lurfaoe and solidity of a sphere whose diameter is 30 45. I4S. ur ! feet Ans. 14137-2 cub. ft. ; 2827-44 sq. ft Find the surface of a spherical segment whose height ia 4 inches, the diameter of the sphere being 6 feet. Ans. 904-78 sq. ft Find the lolid contents of a hyperboloid, whofe middle di»> meter is 30, end diameter 40, and altitude 24. Ans. 25132'& PROBLEMS FOR PRACTICB. 05 148 Find the loliditj and surface of a regular pyramid, whoM base is a square, each side being 6 feet, the apothem or pv- pendicular on the side of the pyramid being 40 feet. Ans. Surface =616 sq. ft. 149. Find the volume of a middle zone of a sphere, whose height to 8 >'eet, the diameters of the ends being 6 feet. Ans. 4f)4 278. 150. Find the contents of a regular hexagonal frustum, whose alt^ tude is 6 feet, the side of the greater end 18 inches, and qT the smaller end 12 inches. Ans. 24 6817 cub. ft. Problems for Practice. 1. Find the area of a square, whose side is 13 chains. 2. Find the side of a square, whose area is 3 acres, 14 per., 18 yards. 3. What is the area of a square, whose diagonal is 260 yards ? 4. The area of a square is 7 acres, I rood, 30 per., required tke length of its diagonal in yards. 6. (a) Find the area of a rectangle, whose length Is 700 and breadth 600 links. (6) Find the area of a parallelogram, whose base is 600 aai perpendicular 250 yards. Answer in acres, roods, &c. 6. (a) The area of a rectangle is 700, its breadth is 35, what is Ub length ? {b) The area of a parallelogram is 4 acres, 3 roods, 16 yards, its breadth is 120 yards, what is ita length 7 T. (a) The area of a rectangle is 17 acres, I rood, 16 per., ita length is 1600 links, what is its breadth ? (6) The area of a parallelogram is 1600. its length is 240, whct is the perpendicular distance between its sides? 8. Find the area of a rectangle, whose diagonal is 500 links, and :?' breadth 300 links. 9. Find the bypothenuse of a right angled triangle, whose bale is 75 yards and perpendicular 48 yards. 10. The hypothenuse of a right angled triangle is 600, the perpei- dicular is 230, what is the base ? il. The hypothenuse of a right angled triangle is 73, the base is 29, what is the perpendicular? 66 MENSURATION. ••;'*'■ fi V It r i ■ill i i! S-' ." n. r-:! si In k 15 16 12. In a, right angled triangle the hypothenuse is 50, the perpen- dicular 20, what are the segments into which a perpendicu- lar from the right angle cuts the hypothenuse ? 13 In a right angled triangle the hypothenuse is 40, the base 16, into what two segments does a perpendicular frum the right angle cut the hypothenuse ? 14. In a right angled triangle the segments into which a perpen- dicular from the right angle cuts the hypothenuse are 40 and 30, what is the perpendicular distance of the right angle from the hypothenuse? Find the area of a triangle, whose base is 750 and altitude 340 links. The area of a triangle is IT acres, 4 per., 16 yards, the altitude is 570 ytvrds, what is the length of the base ? 17. The area of a triangle is 640, the base is 120, what is the alti- tude? 18. Find the area of a triangle, whose three sides are respectively 640, 320, 480 links. What is the area of an equilateral triangle, whose base is 160 yards long ? Find the area of a trapezoid, whose parallel sides are 500 and 300 links, ihe perpendicular distance between them being 120 links. 21. Find the area of a quadrilateral field, whose diagonal is 420 yards, the perpend' culars or the diagonal from the opposite angle being 70 and 130 yards. Find the area of a quadrilateral which may be inscribed in a circle, its four sides being respectively 80, 90, 100 and 120 yards long. Find the area of a regular octaguj, whose side is 13 feet. (See Table I. for apothem, page 72). The area of a regular hep*.iigon is 5 acres, 1 rood, 27J per., find the length of a side. (iSec^ Table I., page 72). 25. The area of a regular polygon is 42784 square feet, each side is 20 feet long and the apothera 37-32 feet, how many sides has the polygon ? , ... 26. Find the circumforence of a circle whose diameter is 12|. 27. Find the diameter of a circle whose circumference is 180 yds. 19 ^0 22 23, 24 . ^1 ■ ■ PROBLEMS 70R PRAGTICB. 67 SA 28. Find the area of a circle whose diameter is 14 and circumfer- ence 43 9824 feet. 29. liequired the diameter of a circle whose area is 490 875 square yards, and circumference 78 54 yards. 30. Required the circumference of a circle, whose area is 1256-64 square feet and diameter 2040 feet. 31. Find the area of a circle, whose diameter is 840 links. 32. The area of a circle is 15 acres, 2 roods, 16 per., 20 yards, what is its diameter in links? 33. Find the area of a circle whose diameter is 220 yards. 34. Find the area of a circle whose circumference is 330 links. 35. What is the area of a circular annulus, the diameters of the concentric circles being 70 and 50 feet ? 36. What is the area of a circular annulus, the circumferences of containing circles being 80 and 280 ? 37. Required the area of a circular annulus, the containing circles having circumferenceij 251328 and 439-824 links, and diame- ters 80 and 140 li iks. 38. The diameter of a circle is 60 feet, what is the length of an arc of 87^** of the circumference? 39. What is the length of the chord of a circle, the diameter of the circle being 61-61 16 and apothem 25 ? 40. Find the apothem on the <'hord of a circle, whose diameter is 244, the length of the chord being 24. 41. Find the chord of half the arc of a circle, whose radius is 18-75, the height of the whole arc being 6. 42. Find the radius of a circle, the lieight of the arc being 4, and the chord of half the arc being 20. 43. Find the area of the segment of a circle, whose diameter is 45-3 feet, the arc containing 2409, the chord being 40 feet and apothem 10 feet. ^ 44. Find the area of a sector of a circle, whose radius is 50 yards, the length of the ai c being 90 yards. 45. Find the area of the sector of a circle, whose radius is 80 feet, the circular arc containing 72 degrees. 46. Find the area of a lune, whose common chord is 40 feet, the length of the outer arc is 94-876 and of the inner one 60 feet j the apothem of the outer smaller circle being 12-65 feet, and rM^-- MENSURATION. -m fi" 11^ ill w %l yi ui 'M m 68 « its diameter 45*3 feet, the apo.them of the larger circle being 48 feet and its diameter 200 feet. ,^ ;; ,, ^ 47. Find the circumference of an ellipse whose axes are 16 and 28. 48. Find the area of an ellipse whose diameters are 20 and 14. 49. Required the ordinate of an ellipse, whose diameters are 90 and 10, and one absciss 24. 50. Find the abscisses of an ellipse, whose axes are 60 and 80, and an ordinate 30. 51. What is the transverse axis of an ellipse, whose conjugat* axis is 70, an ordinate 28, and one absciss 168 ? 52. What is the conjugate axis of an ellipse, whose major axis U 70, an ordinate 20, and the smaller absciss 147 53. Find the parameter of a parabola, the ordinate and one absclM being 12 and 28. 54. Two abscisses of a parabola are 9 and 16, the ordinate of the former is 6, find that of the latter. 55. An absciss of a parabola is 32 and its ordinate 24, a second ordinate is 18 ; what is its absciss 7 56. Find the length of the arc of a parabola, whose absciss and ordinate are 3 and 5. 57. Find the area of a parabola, whose base and height are 20 and 28. 58. Find the area of a parabola zone, whose bases are 7 and 10 and height 6 feet. 59. In an hyperbola the axes are 90 and 45, the less absciss is 30 ; find the ordinate. 60. The axes are 15 and 7^, the ordinate 5 ; what are the abscisses of the hyperbola ? 61. What is the solidity of a tetrahedron, whose edge is 5 inches ? 62. In an hyperbola, the transverse axis is 2.5, the less absciss 8}, and its ordinate 10; required the conjugate axis. 63. The conjugate axis is 31^, the smaller absciss 12, the ordinate 21 ; what is the transverse axis of the hyperbola? 64. What is the area of an hyperbola, whose axes are 15 and 9, the smaller absciss being 5 7 65. What is the surface of a regular triangular pyramid, whoM edge is 7 feet 7 66. What is the aggregate surface of a cube whose edge is 8} feet 7 I < I f PROBLEMS VOR PRAOTIOB. m „ 6T. What arts the solid oontents of a hexahedron whose edge If U ^:^j. . ■• ■ feet long? ,.,,•::.-.„,, 68. Find the surface of an octahedron whose edge is 4} feet? 60. Required the eubic contents of an octahedron whose edge it 15 inches. '\ 70. The edge of a dodecahedron is 6 inohos, what is its entiN surface? 71. Find the yolume of a dodecahedron whose edge is If feet 72. Required the surface of an icosahedron whose edge is 10} inches. 73. What are the cubic contents of an icosahedron, whose edge is 20 inches ? 74. (a) What is the volume of a right rectangular paraIlelo> piped, whose length is 20 feet, breadth 4} feet, and height 18 feet? (b) What is the volume of an oblique triangular prism, the edges of the end being 7, 9 and II inches, and the length of the prism 45 inches? («) What is the volume of a hexagonal prism, whose length is 22 feet, each edge of the end being 20 inches. 75. (a) Find the surface of a right rectangular parallelopipedi whose base is 16 inches bj 9 inches, the height of the solid being 4 feet. (6) Find the surface of a right octagonal prism, whose height is 12 feet, each edge of the end being 2} feet. 76. Required the solidity of the frustum of a cone, whose height is ^ 5 feet and end diameters 4 and 2 feet. 77. Find the surface of an oblique pentagonal prism, whose length is 4^ feet and edge 3 feet, the perimeter of a section perpen- ; dicular to one of the lateral edges being 16 feet. : ' 78. (a) Find the volume of a regular pyramid, whose height is 10 feet, the base being a heptagon, whose side is 2 feet. (6.) Find the volume of a regular cone, whose base has a dia- meter of 7 feet, the height of the cone being 9 feet. 79. (o) Find the entire surface of a regular octagonal pyramid, whose height is 11 feet, each edge of the base being 5 feet. (^•) Find the entire surface of a right cone, whose height is 14 feet, the diameter of the base being 7} feet. '1- 70 \S:''^i;- MENSURATION. !-:; -'%t 80 lil t ■ f: i 81. 82 83 Find the solidity of a frustum of a pyramid, whope height is 5 feet, the areas of the two ends heing 12 and 18 square feet. Find the volume of the frustum of a right pentagonal pyr*- mid, the upper end edges being 3^ feet and the lower 5 feet each, and the height of the frustum 7 feet. Find the volume of the frustum of a hexagonal pyramid, the edge of the bottom being 4 feet and of the top 2j feet, while the height of the frustum is 6 feet. What are the solid contents of the frustum of a cone, whose height is 10 feet, the end diameters being 5 and 3 feet ? 84. What is the whole surface of the frustum of a cone, whose end diameters are 4 and 8 feet, the slant side being 6} ft. long? 85. Find the volume of a wedge, whose edge is 13 inches and back 10 inches long, the breadth of the back being 4J inches and the length of the wedge 2 feet. 86. Find the solidity of a spherical segment, whose height is 6feet, the radius of the base being 2 feet. What are the solid contents of a spherical segment, whose height is 7 inches, the diameter of the sphere being ;,10 inches? Find the convex surface of a spherical segment, whose height is 10 inches, the diameter of the sphere bein^ 4 feet, [2 inches. 89. Find the volume of a sphere whose diameter is 8i feet. 90. Find the surface of a sphere whose diameter is 7924 miles. 91. Fin J. the volume of a spherical zone, whose height is 2 feet, the radii of the ends being 3 feet and 4 feet. 92. Find the volume of the middle zone of a sphere, the height of the zone being 4 feet and the diameter of either end 3 feet. 93. What is the volume of the middle zone of a sphere, the height of the zone being 6 feet and the diameter of the sphere 10 feet? Find the convex surface of the middle zone of a sphere, the height of the zone being 7 feet and the diameter of the sphere 20 feet. Find the volume of a paraboloid, whose height is 10 feet and diameter 8 feet. 87. 88 94. 95. 96 97 98 99 IC IC 1( 1( II I \ K h PROBLBMS OF PRACTIOB. 71 11 « 96. Find the volume of the frustum of a paraboloid, whose height is 5 feet, the areas of the ends being 7 and 9 square feet. 97. What is the volume of the frustum of «, paraboloid, whose height is 8 feet and end diameters 10 and 4 feet? 98. Find the volume of an oblate spheroid, whose diameters are IS and 17 feet. 99. Find the volume of a prolate spheroid, whoso axes are 7 and 11 feet. ' 100. What is the volume of a circular segment of an oblate spheroid, the axes being 18 and 12 feat, and the height 4 feet? 101. What is the volume of a circular segment of a prolate spheroid, the axes being 10 and 15 and the height 6 feet. 102. Find the volume of an elliptical segment of an oblate spheroid, whose height is 4 feet, the axes being 12 and 16 feet. 103. Find the volume of an elliptical segment of a prolate spheroid, whose height is 6 foet, the axes being 18 and 20 feet. 104. Find the solid contents of a circular middle frustum of a pro- late spheroid, whose conjugate axis is 12 feet, the diameter of the frustum being 6 feet and its height 7 feet. 106. Find the volume of a circular middle frustum of an oblate spheroid, the diameter of the frustum being 6 feet, the transverse axis 20 feet, and the length of the frustum 8 feet. 106. Find the cubic contents of an elliptical middle segment of a spheroid, whose axes are 16 and 24, the length of the frus- tum being 12 feet, and the greater and less diameters of either end 6 and 4 feet. 107. Find the solidity of an hyperboloid, whose base has a dia- meter of 10 feet, the diameter half way between the base and vertex is 6 feet, and the height of the hyperboloid 8 feet. 108. Find the solidity of a frustum of an hyperboloid, whose height is 6 feet, the radii of the ends being 3 and 7 feet, and the diameter half way between the ends 12 feet. 72 HBNSURATIOZr. nt^ Tablb I., ^V^'-:^.'W Showino Apothbh and Abba or Poltooxs. -■-•''^ i in M m i^ t ; Name of Polygon. No. of Sides. Apothem when gide= 1. Area when side = . THanirle < 3 4 5 6 7 8 9 10 11 12 0-2886751 0-5 0-6881910 0-8660254 1-0382607 1-2071068 1-3737387 1-5388418 1-7028436 1-8660254 0-4330127 1* 1-7204774 2-5980762 3-6339124 4-8284271 6-818242 7-6942088 9-3656399 11-1961524 * Square Pentasron Hexasron HeDtasron Octaeron NonaGTon Decazoa Undecacron Dodecacron J ^ -.! ?■ - , ■ , i: , ^^iifi--^-^ WS-: •■ i4t ' ' i Table ii. ■y A _, ; f A gallon of water weighs 10 lbs. avoir. A cubic foot of water weighs 62i lbs. ^ 1000 oz. A pail of water = 3} gallons 3^25 lbs. A gallon is equal to 277274 cubic inches. Table iir. LAND MBASUBB. " '. 7-92 inches = 1 link. 100 links = 1 chain. 80 chains = 1 mile. if/* 10000 square links = 1 square chain. 10 square chains =or 100,000 square links = 1 acre. 1 chain — 4 rods. 1 acre— 160 square rods = 4840 square yards. NoTB.— If we desire to compute area of polygon by tabular area, w« must remember that similar polygons are to each other as squares of homologous sides; hence 1>: side^i: : tabular area : required area. f ; i t' I,'. i- ;•' n I, w« BB of