830 COROLLARY 1. The surface of a sphere is equivalent to four great circles.

For SAD × 2 TR = 2 R × 2 TR = 4 TR2.

831 COROLLARY 2. The areas of the surfaces of two spheres are to each other as the squares of their radii, or as the squares of their diameters.

832 COROLLARY 3. The area of a zone is equal to the product of its altitude by the circumference of a great circle.

For the surface generated by any arc BC is a zone the area of which by the proof of the theorem equals EF × 2 TR, in which EF is the altitude of the zone (§ 815), and 2 TR is the circumference of a great circle.

833 COROLLARY 4. Zones on the same sphere or equal spheres are to each other as their altitudes.

For denote the areas of two zones by Z and Z', and their altitudes by H and H'.

Then, Z: Z' = H × 2 πR : H' × 2 πR = H: H'.

834 COROLLARY 5. A zone of one base is equivalent to the circle whose radius is the chord of the generating arc.

For zone AB = AE × 2πR=TAE × AD=TAB2. § 365, Case 3

835 COROLLARY 6. The area of a lune =

For a spherical degree

TR2A

§ 822

and the area of a lune = 2 A spherical degrees

SPHERICAL VOLUMES

DEFINITIONS

837 A spherical segment is a portion of a sphere included between two parallel planes.

The bases of a spherical segment are the sections of the sphere made by the parallel planes.

The altitude of a spherical segment is the perpendicular between the bases.

If one of the parallel planes is tangent to the sphere, the segment formed is called a spherical segment of one base.

E

F

B

Let ACDB be a semicircle, CE and DF perpendicular to AB the diameter. If the semicircle is revolved about AB as an axis, the solid generated by the figure CEFD is a spherical segment, whose altitude is EF, and whose bases are the circles generated by CE and DF.

The solid generated by the figure BDF is a spherical seg

ment of one base.

If DF is a radius, the spherical segment generated by BDF is half a sphere.

838 A spherical sector is the portion of a sphere generated by any sector of a semicircle when the semicircle is revolved about its diameter as an axis.

The base of a spherical sector is the zone generated by the arc of the circular sector.

If the base of a spherical sector is a zone of one base, the spherical sector is called a spherical cone.

D

C

A

B

Thus, when the semicircle ACDB revolves about AB as an axis, the solid generated by the circular sector COD is a spherical sector, whose base is the zone generated by the arc CD.

The solid generated by the sector AOC is a spherical cone.

839 A spherical pyramid is the portion of a sphere bounded by a spherical polygon and the planes of the great circles forming the sides of the polygon.

The vertex of a spherical pyramid is the center of the sphere. The base of a spherical pyramid is the spherical polygon which forms a part of its boundary.

Thus, O-ABCD is a spherical pyramid, whose vertex is 0, and whose base is the spherical polygon ABCD.

840 A spherical wedge, or ungula, is the portion of a sphere bounded by a lune and two great semicircles.

Thus, EOFHK is a spherical wedge.

PROPOSITION XXVIII. THEOREM

841 The volume of a sphere is equal to the product of the area of its surface by one third of its radius.

HYPOTHESIS. volume V.

O is the center of a sphere whose radius is R, surface S, and

CONCLUSION. V=SX R.

PROOF

Circumscribe about the sphere any polyedron; denote its surface by S' and its volume by V'.

Straight lines drawn from O to each of the vertices of the polyedron divide the polyedron into as many pyramids as it has faces, whose bases are the faces of the polyedron, whose common vertex is O, and whose common altitude is R. The volume of each of these pyramids is equal to the area of its base multiplied by R.

§ 667 Therefore, the combined volume of all these pyramids, which is the volume of the circumscribed polyedron, is equal to the area of the surface of the polyedron multiplied by R.

That is, V'S' x R, no matter how many faces the circumscribed polyedron may have.

By passing planes tangent to the sphere at the points in which the lateral edges of the pyramids pierce the surface of the sphere, a part of the polyedron will be cut away, and a new

polyedron, having more faces than the first, will be circumscribed about the sphere, whose volume will be less than the volume of the first polyedron.

Ax. 12

By joining O to each of the vertices of this new polyedron, the polyedron will be divided into a new series of pyramids, greater in number than the first series, but whose combined volume is likewise equal to the area of the surface of the new polyedron multiplied by R.

By repeating indefinitely this process of circumscribing new polyedrons about the sphere, the volume of each will be less than the volume of the preceding one (Ax. 12), and hence V' will be a decreasing variable approaching V as a limit.

Since in the equation V'S' x R, V' is a decreasing variable, and R is a constant, therefore S' is a decreasing variable approaching S as a limit.

Since S' approaches S as a limit,

843 COROLLARY 2. The volumes of two spheres are to each other as the cubes of their radii.

844 COROLLARY 3. The volume of a spherical pyramid is equal to the product of the area of its base by one third of the radius of the sphere.