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776. Cor. In a spherical segment of one base, r1=0.

EXERCISES

Ex. 1234. Find the volume of a spherical segment, the radii of whose bases are 4 and 5 and whose altitude is 1.

Ex. 1235. The volumes of two spheres are to each other as 8 to 125. Find the ratio of their radii.

Ex. 1236. The volumes of two spheres are to each other as 125 to 216. Find the ratio of their surfaces.

Ex. 1237. Find the radius of a sphere whose surface is equivalent to the sum of the surfaces of two spheres whose radii are 3 and 4 respectively.

Ex. 1238. Find the volume of a spherical shell whose exterior radius is 13 and whose thickness is 8.

Ex. 1239. Find the radius of a sphere equivalent to the spherical shell in the preceding exercise.

Ex. 1240. Find the radius of a sphere equivalent to a cube whose edge is equal to a.

Ex. 1241. A cylindrical vessel, 4 in. in diameter, is partly filled with water. Upon immersing a ball the surface of the water rises 1 in. Find the diameter of the ball.

Ex. 1242. A sphere whose radius is 2 in. weighs 32 oz. Find the weight of a sphere of the same material whose radius is 3 in.

Ex. 1243. Find the volume of a spherical pyramid whose base is an equilateral triangle with its angles equal to 80°, if the radius of the sphere is equal to 10.

Ex. 1244. A square whose side is 4 revolves about one of its diagonals. Find the surface and the volume of the generated solid.

Ex. 1245. Find the volume of a spherical segment of we base, if its curved surface is 20 and its altitude is 2.

Ex. 1246. Find the radius of a sphere whose surface is equivalent to the entire surface of a cube whose edge is equal to 4.

Ex. 1247. The edge of a cube is 10 in. Find the diameter of the circumscribed sphere.

Ex. 1248. A lune whose angle is equal to 40° is equivalent to a zone on the same sphere. Find the ratio of the altitude of the zone to the radius of the sphere.

Ex. 1249. The diedral angles of a spherical pyramid of six sides are 140°. Find the volume of the pyramid if the radius is equal to 10.

Ex. 1250. Through a sphere whose diameter is 10 m. a cylindrical hole of 5 m. diameter is bored. Find the volume of the solid if the axis of the cylinder passes through the center of the sphere.

Ex. 1251. The surface of a sphere is equivalent to the lateral surface of the circumscribed cylinder.

Ex. 1252. Two bi-rectangular spherical triangles are equal if the oblique angles are equal.

Ex. 1253. Find the ratio of a sphere to its circumscribed cube.

Ex. 1254. The area of a zone on a sphere is 20, its altitude 4. Find the radius of the sphere.

Ex. 1255. If the diagonals of a spherical quadrilateral bisect each other, the opposite sides are equal.

Ex. 1256. The radius of a sphere is 9 in. Find the volume of a spherical wedge whose angle is equal to 60°.

Ex. 1257. Find the radius of a sphere equivalent to a cone of revolution, the radius of whose base is equal to r and whose altitude is equal to h.

Ex. 1258. The area of a zone is equal to A, its altitude is equal to h. Find the radius of the sphere.

Ex. 1259. The volume of a sphere is numerically equal to one-half its surface. Find the radius.

Ex. 1260. The volume of a cylinder of revolution is equal to one-half the product of its lateral surface by the radius of its base.

Ex. 1261. How many square miles of the surface of the earth can be seen from a point 1000 miles above the surface, if the earth is supposed to be a perfect sphere whose radius is equal to 4000 miles ?

Ex. 1262. If from a point without a sphere a tangent and a secant be drawn, the tangent is the mean proportional between the secant and its external segment.

Ex. 1263. If through the line of intersection of two spheres a plane be passed, tangents from a point of the plane to the spheres are equal.

Ex. 1264. The radius of a sphere is r, the area of a small circle a. Find its distance from the center.

Ex. 1265. The volume of a sphere is V. Find the surface of an equilateral spherical triangle whose angle is equal to 100°.

INDEX OF DEFINITIONS

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PAGE
Alternation . . . . . . . 120 Angle, tetraedral . . . . . 262
Altitude of cone . . . . . 314 16 triedral . . . . .

of cylinder . . . . 306 66 vertex of . . . . .
of frustum of pyramid 288

vertical . . . . . .
of parallelogram . , 50 | Angles, adjacent . . . . . 5
of prism . .. . . 271 alternate exterior..
of pyramid . . . . 287 66 alternate interior ..

of spherical segment. 361 " complementary . . .
66 of trapezoid.... 80

corresponding . ..
66 of triangle . . . . 12

exterior. . . . .
of zone . . . . .

66 interior . . . . .
Analysis of theorems . . . 61

of polygon. . . . . 11
o of problems . . . 107 66 supplementary . . . 5
66 algebraical. ...

vertical. ... 5
Angle . . . . . . . . . 3 Antecedents . . . . . . 118

acute . . . . . . . Apothem. .......

central . . . . . . 74 Arc . . . . . . . . .
66 diedral . . . . . . Area . . . . . . . . .
" exterior in triangle . . 11 | Arms of isosceles triangle . . 11

inscribed in circle .. " of right triangles . . . 12
inscribed in segment of Axiom . . . . . . . . 7

circle . . . . . . Axis of circular cone. . . . 314
u oblique . . . . . . ut of circular cylinder . . 306

obtuse . . . . . . 4 66 of regular pyramid . . 287
of lune . . . . .
of two curves . . . . 335 Base of isosceles triangle . . 11
polyedral . . . . . 262 6 of pyramid . . . . . 286
reflex . . . . ... 4 of spherical pyramid . . 320

right . . . . . . . 4 os of spherical sector. . . 360
" spherical . . . . . 335 " of triangle . . . . . 11
straight . . . . . . 4 " of cylinder . . . . . 305

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Bases of frustum of cone . . 315 Cube . . . . . . . .

272
66 of frustum of pyramid . 288 Cylinder . . . . . . . . 305
or of parallelogram · · · 50

circular ..
of prism . . . . . . 271

oblique . . . .

• 306
66 of spherical segment. . 397 |

of revolution .. 306
66 of trapezoid . . . . 50 66 right . . . . . 305
66 of zone . . . . . . 358

right section of . . 307
Bisector . . . . . . . . 4 Cylindrical surface .... 305
" of triangle . . . . 12

oo directrix of . 305

66 element of . 305
Center of circle ..... 73

66 generatrix of 305
66 of regular polygon. . 213
66 of sphere . . . . .

Decagon . . . . . . . . 57
Chord . . . . . . . . .

Degree . . . . . . . . 5
Circle . . . . . . . .

Determined plane. ....
Circles, concentric .... 74

Diagonal of polygon . ..
66 escribed . . . .

Diameter of circle ...
Circumference . . . . .

" of sphere... 327
Commensurable i ...

Diedral angle .....
Complement . . . . .

6 edge of..

.251
Composition . . . . . . 121

66 66 faces of . . . 251
and division.

" " plane angle of . 252
Conclusion · · · · · · ·

Dimensions . . . . . . . 11
Cone · · · · · · · 313

Distance, from point to line . 41
u circular .....

66 from point to plane . 245
66 lateral surface of . . . 313

" on surface of sphere 329
oblique . . . . . . 314

Division . . . . . . . .
66 of revolution . . . 314

u interior . . . . . 125
66 right . . . . . . 314

66 exterior . . . . . 126
" vertex of . . . . . . 313

Dodecaedron ....
Conical surface . . . . . 313

Dodecagon · · · · · · · 57
66 directrix of . 313

element of . . 313
generatrix of . 313 Edge of diedral angle... 251
nappes of . . 313 Edges of polyedral angle .. 262

vertex of . . 313 1 " of polyedron.... 270
Consequents . . . . . . 118 Element of conical surface. . 314
Constant . . . . . . . . 91 " of cylindrical surface 305
Continued proportion ... 118 Equal figures . . . . . . 2
Converse of a theorem ... 8 Equivalent figures . . . . 164
Corollary . . . . . . . 71

66 solids . . . . . 276

122

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Extreme and mean ratio . 151 Major arc of circle . . . . 73
Extremes . . . . . . . 118 Maximum . . . . . . .

Mean proportional . . . . . 118
Faces of diedral angle . . . 251 Means . . . . . . . . . . 118

" of polyedral angle . . 262 Median of triangle . ...

66 of polyedron . . . . Minimum . . . . . . . . 225
Figure, geometrical . . . .

Minor arc of circle. . . . . 73
" plane . . . . .

66 rectilinear . . . Nappes of cone ...... 313
Foot of perpendicular...

Numerical measure .... 90
Fourth proportional . . . . 118
Frustum of cone . . . . . 315 Octaedron. . . . . . . . 270

Octagon ......57
Generatrix of conical surface . 313
os of cylindrical sur-

Parallel lines. ......
face . . . . 305

Parallelogram .....
Geometry . . . . . . . 2

Parallelopiped .....
" Plane . . . . . 2

" right..
Solid .. .

rectangular . . 272
Harmonic division ....

Parallel planes .....

Pentagon . . . . . . . .
Heptagon . . . . . . . 57

Perimeter .......
Hexagon . . . . . .

Perpendicular .....
Hexaedron. . . . . .

Plane . . . . . . . .
Homologous . . . . .

Point .........
Hypotenuse . . . . . . . 12

Polar distance of circle ...
Hypothesis . . . . . . ..

" triangle . .. .
Icosaedron. . . . . . . 270 Pole . . . . . . . . .
Inclination of line to plane

. 11
Polygon
.259

.......
Incommensurable . . . . . 90

66 angles of ....
Inversion . . . . . . . 121

circumscribed . .
Isoperimetric figures . . . . 225

convex . . . . .

diagonal of ...
Limit . . . . . . . . . 91

equiangular ...
Line . . . . . . . . . . 2

equilateral . . .
Line, broken . . . . . . 2

inscribed ...
curved . . . . . . 2 " regular . ...
66 of centers . . . . . 87 66 spherical ...
Locus . . . . . . . . . 111 Polyedral angle . . . . . . 262
Lune . . . . . . . . . 350 Polyedron . . . . . . . . 270

66 angle of . . . . . . 350 | 66 convex . . . . . 270

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