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776. COR. In a spherical segment of one base, rı = 0.

hm2 nh3 .. V=

2

6

+

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 respec. tively.

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 one 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
262
262

3
11

5
17
17

5
17
17
17
11
5

5
118
195

73
194
11
12

7
314
306
287

PAGE
Alternation.

120 Angle, tetraedral
Altitude of cone

314

triedral
of cylinder

306

vertex of
of frustum of pyramid 288

vertical.
of parallelogram 50 Angles, adjacent
of prism .

271

alternate exterior
of pyramid

. 287

alternate interior
of spherical segment. 361

complementary
of trapezoid.

80

corresponding
of triangle

12

exterior.
358

interior.
Analysis of theorems

61

of polygon.
of problems

107

supplementary
algebraical

219

vertical .
Angle .

3 Antecedents
acute.

4 Apothem.
central

74 Arc
diedral

251 Area
exterior in triangle 11 Arms of isosceles triangle
inscribed in circle

95

of right triangles
inscribed in segment of

Axiom
circle

95 Axis of circular cone.
oblique

5

of circular cylinder
obtuse

4 of regular pyramid
of lune
of two curves

335 Base of isosceles triangle
polyedral

262 of pyramid
reflex

4

of spherical pyramid .
right.

4

of spherical sector.
spherical

335

of triangle
straight

4 of cylinder

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350

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PAGE
Bases of frustum of cone . 315 Cube

272
of frustum of pyramid 288 Cylinder .

305
of parallelogram

50
circular

305
of prism . .

271
oblique

306
of spherical segment. 397

of revolution

306
of trapezoid

50
right

305
of zone

358

right section of 307
Bisector
4 Cylindrical surface

305
of triangle

12

directrix of . 805

element of . 305
Center of circle

73

generatrix of 305
of regular polygon

213
of sphere

327
Decagon .

57
Chord.

74
Degree

5
Circle

33, 73
Determined plane.

234
Circles, concentric

74
Diagonal of polygon

11
escribed

106
Diameter of circle

73
Circumference .

33, 73
of sphere

327
Commensurable

90
Diedral angle

251
Complement

5
edge of

251
Composition

121

faces of

251
and division 122

plane angle of 252
Conclusion

8
Dimensions .

11
Cone

313

Distance, from point to line 41
circular

314

from point to plane . 245
lateral surface of

313

on surface of sphere 329
oblique

314
Division.

121
of revolution

314
interior

125
right

314
exterior

126
vertex of

313
Dodecaedron

220
Conical surface

313
Dodecagon

57
directrix of 313
element of . . 313
generatrix of . 313 Edge of diedral angle

251
. 313 | Edges of polyedral angle 262
vertex of 313

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

7
solids.

276

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90

270
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PAGE
Extreme and mean ratio 151 Major arc of circle
Extremes

118 Maximum .

Mean proportional .
Faces of diedral angle

251

Means
of polyedral angle

262

Median of triangle .
of polyedron

270

Minimum . .
Figure, geometrical

2

Minor arc of circle.
plane

2
rectilinear

9

Nappes of cone .
Foot of perpendicular

4

Numerical measure
Fourth proportional

118
Frustum of cone

315

Octaedron.

Octagon
Generatrix of conical surface. 313
of cylindrical sur-

Parallel lines.
face

305

Parallelogram
Geometry

2

Parallelopiped
Plane

2

right
Solid

2

rectangular

Parallel planes
Harmonic division

129

Pentagon
Heptagon

57

Perimeter
Hexagon

57

Perpendicular
Hexaedron

270

Plane
Homologous

13

Point
Hypotenuse .

12

Polar distance of circle
Hypothesis

7

triangle
Icosaedron

270

Pole.
Inclination of line to plane

259 Polygon
Incommensurable.

angles of
Inversion

121

circumscribed
Isoperimetric figures .

225

diagonal of
Limit .

91

equiangular
Line

2

equilateral
Line, broken

2

inscribed
curved

2

regular
of centers

87

spherical
Locus.

111 Polyedral angle
Lune

350 Polyedron .
angle of

350

convex

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272

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236
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4
2, 234

1
330
342
329
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14
76
194
337
. 262

270
270

90

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