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297. COR. 1. The area of a spherical triangle is the same part of the area of the sphere that its spherical excess is of 720°.

298. COR. 2.

E

T=

7200 (4πу2).

The area of a spherical polygon is the area of the sphere that its spherical

same part of the

excess is of 720°.

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299. Notation for Spherical Triangles and Spherical Polygons. In finding the areas of spherical polygons, since only the angles are used, a convenient notation is to denote a triangle by T with a subscript to show its angles. Thus, T(A, B, C) would represent the triangle of angles A, B, and C, and T(97°, 85°, 103°) would represent the triangle having angles of 97°, 85°, and 103°. It is convenient also to denote a polygon of more than three sides by P with a subscript to show its angles.

264. On a sphere of radius 10 ft., find the area of (a) a triangle of angles 84° 30', 111° 28', 35° 17'; (b) a pentagon each angle of which is 120°.

265. If an equiangular triangle is one tenth of the spherical surface, find its angles.

266. A spherical triangle whose angles are in the ratios 1:2:3 is equivalent to a zone of altitude. Find its angles.

3 267. A spherical polygon on a angle 160°, and its area is 44 sq. in.

sphere of radius 6 in. has each How many sides has it?

268. Find the area of T (390, 380, 1790) in terms of r.

269. Find the area of P(168°, 178°, 107°, 123° 30′, 177° 14′, 154° 16′) on a sphere of radius 1 ft.

SECTION IV. VOLUME OF A SPHERE AND ITS PARTS

300. One method of finding the volume of a sphere is to consider it the limit approached by a circumscribed polyhedron as the number of its faces is increased indefinitely. This corresponds to the method used to find the area of a circle. As in dealing with the circle, there must be assumed a statement that the figure approaches a limit as the number of its faces is increased.

If each vertex of a circumscribed polyhedron is joined to the center of the sphere, and planes are passed tangent to the sphere at the points where these lines meet the surface (thus forming, with the intercepted parts of the faces of the original polyhedron, a new polyhedron of a larger number of faces), the area and the volume of the polyhedron approach the area and the volume of the sphere as limits, as the number of faces is increased indefinitely.

301. Theorem XX.

The volume of a sphere is one third the product of its radius and its area.

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FIRST METHOD. The volume of a circumscribed polyhedron is one third its area times the radius (why?). Increase the number of faces indefinitely.

SECOND METHOD.

Show that a sphere of radius r, and a cylinder of radius r and altitude 2r that has been hollowed out from each base to the center in the form of right circular cones of radius r and altitude r, are Cavalieri

bodies, since the section of each at a distance d from the center is Tr2 πd2. Then find the volume of the hol

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lowed cylinder.

302. Solids Extending from the Center of the Sphere to its Surface. The first method of proof is applicable to all solids that extend from the center of the sphere to some part of its surface, if they are bounded laterally by planes or conical surfaces, for all such solids can be considered as limits of the sum of an indefinitely large number of pyramids with vertices at the center of the sphere, and bases tangent to the sphere.

303. Spherical Cones and Spherical Sectors. That part of a sphere included in a conical space whose vertex is at the center of the sphere is called a spherical cone (as 0-B1).

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It could be generated by revolving a plane sector about one of its bounding radii as an axis, so it is considered as one kind of spherical sector. That part of a sphere generated by revolving a plane sector about a diameter is called a spherical sector (as 0- B2); if it revolves about a diameter other than one of its own boundaries, it is evidently the

difference of spherical cones. The base of any kind of spherical sector is evidently a zone. The spherical cone 0-B1 is cut out of sphere O by the conical space O-CD.

*304. The volume of a spherical cone or a spherical sector is one third its base times the radius of the sphere. V = } πr2h

(where h is the altitude of the zone).

270. Express the formula for the volume of a spherical cone in terms of the distance of the plane cutting off the zone from the center of the sphere, instead of in terms of h.

271. Express the formula for the volume of a spherical cone in terms of the radius of the circle instead of in terms of h.

272. Find the volume of a spherical sector in a sphere of radius 12 in. if the radii of the two circles of its base are 6 in. and 8 in.

B

305. Spherical Pyramids. That part of a sphere included in a pyramidal space whose vertex is at the center of the sphere is called a spherical pyramid. The base of a spherical pyramid is evidently a spherical polygon. The spherical pyramid, like any other pyramid, can be triangular, quadrangular, etc. O-ABC is a triangular spherical pyramid.

C

1

0

*306. The volume of a spherical pyramid is one third its base times the radius of the sphere.

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where E is the spherical excess of the base.

273. State the volume of a spherical pyramid in form corresponding to that used for the area of a spherical triangle in § 297.

307. The Spherical Wedge. That part of a sphere between two great semicircles is called a spherical wedge or an ungula. Its curved surface is evidently a lune.

*308. The volume of a spherical wedge is one third its base times the radius of the sphere.

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where A is the angle between the circles.

274. State the volume of a spherical wedge in form corresponding to that used for a lune in § 295.

309. Spherical Segments. The part of a sphere that lies between parallel planes is called a spherical segment. The planes can both be secant planes, one can be a tangent plane, or, in the limiting case of the whole sphere, both can be tangent planes. A spherical segment will have one base or two bases according as it is included between a secant plane and a tangent plane, or between two secant planes. The spherical surface of a spherical segment is a zone.

310. Volume of a Spherical Segment. The second method of proof used for finding the volume of a sphere is applicable also to spherical segments, for the spherical segment and the corresponding part of the hollowed out cylinder are Cavalieri bodies, and the volume of the hollowed cylinder can be found by the formulas for the cylinder and the cone. The volume of a spherical segment of one

Th2

base is expressed by the formula v= (3r - h), and

3

that of two bases is expressed by the formula

V =

πh (3 r2 + 3 r22 + h2),

πη
6

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