But, however great the number of pyramids,

Hence, § 222,

v' = s' × { R.

V=SX R.

Q.E.D.

744. Formula: V = S×} R = { πR3 = { πD3.

745. Cor. I. Let R, R' denote the radii, D, D' the diameters, and V, v' the volumes, respectively, of two spheres.

Then, § 744, V=TR3, and v' = {πR";

13

V: V' = { πR3 : — πR13 = R3 : R13 = D3 : D'3 ;

that is, the volumes of two spheres are to each other as the cubes of their radii, or as the cubes of their diameters.

746. Cor. II. The volume of a spherical pyramid is equal to the product of its base by one third of the radius of the sphere.

747. Cor. III. The volume of a spherical sector is equal to the product of the zone which forms its base by one third of the radius of the sphere.

748. Formula: Let R denote the radius of a sphere, C the circumference of a great circle, v the volume of a spherical sector, and H and Z the altitude and area, respectively, of the corresponding zone.

Then, since C = 2πR, and Z = 2πRH, V = } πR2H.

Proposition XXVIII

749. Draw a semicircle; to the extremities of any arc of the semicircumference draw radii, and from their extremities draw lines perpendicular to the diameter. If this figure is revolved about the diameter as an axis, what kind of a solid is generated by the part included between the perpendiculars and the given arc? Between the radii and the given arc? Between each radius and the perpendicular from its extremity? Find an expression for the volume of the spherical segment in terms of the spherical sector and the two cones.

Theorem. The volume of a spherical segment is equal to one half the product of its altitude by the sum of its bases, plus the volume of a sphere of which that altitude is a diameter.

Ex. 881. What is the volume of a sphere whose radius is 9 in. ?

Ex. 882. The circumference of a great circle of a sphere is 36 ft. What is the area of the surface of the sphere ?

Ex. 883. The diameter of a sphere is 13dm. How many cubic decimeters does it contain ?

What is its radius ?

Ex. 884. The volume of a sphere is 1870cu m Ex. 885. The area of the surface of a sphere is 69 sq. ft. What is its diameter ?

Ex. 886. The edge of a cube is 16cm. What is the volume of the circumscribed sphere?

Ex. 887. If the sides of a spherical triangle are 75o, 93°, and 110°, what is the spherical excess of its polar triangle?

Ex. 888. The angles of a spherical triangle are 98°, 110°, and 160°. What is the area of a symmetrical triangle on the same sphere, the radius being 12m ?

Ex. 889. Find the volume of a triangular spherical pyramid, the angles of the base being 58°, 116°, and 145°, and the diameter of the sphere being 20 in.

Ex. 890. What is the area of a lune whose angle is 60° on the surface of a sphere whose radius is 6 in. ?

Ex. 891. What is the area of a zone whose altitude is 3dm on the surface of a sphere whose radius is 8dm?

Ex. 892. What is the volume of a spherical sector whose altitude is 3.5m, if the radius of the sphere is 11m?

Ex. 893. What is the volume of a spherical wedge whose angle is 72°, the volume of the sphere being 1728 cu. in. ?

Ex. 894. What is the area of a zone of one base, if the chord of its generating arc is 13dm ?

Ex. 895. The area of a zone of a sphere 20dm in diameter is 150sq dm ̧ What is the altitude of the zone ?

Ex. 896. The angles of the base of a triangular spherical pyramid are 90°, 121o, and 135°. What is the volume of the pyramid, the volume of the sphere being 194 cu. in. ?

Ex. 897. Spherical polygons are to each other as their spherical excesses. Ex. 898. The base of a spherical pyramid is a trirectangular triangle. What part of the sphere is the pyramid ?

Ex. 899. The surface of a sphere is equivalent to the lateral surface of the circumscribing cylinder.

Ex. 900. What is the spherical excess of a triangle whose area is 261.8 sq. in., if the radius of the sphere is 10 in. ?

Ex. 901. A lune and a trirectangular spherical triangle on the same sphere are to each other as the angle of the lune is to an angle of 45°. Ex. 902. Trirectangular triangles on equal spheres are equal.

Ex. 903. The diameters of two spheres are 12 in. and 14 in. respectively. What is the ratio of their surfaces ? What is the ratio of their volumes?

Ex. 904. The areas of the surfaces of two spheres are as 144 to 24. What is the ratio of their diameters ? What is the ratio of their volumes ?

Ex. 905. The diameters of the sun and earth are in the ratio of 109: 1., What is the ratio of their volumes ?

Ex. 906. How many quarts of water will a hemispherical kettle hold, if its inside diameter is 12 in. ?

Ex. 907. If lines are drawn from any point in the surface of a sphere to the ends of a diameter, they are perpendicular to each other.

Ex. 908. What is the circumference of a small circle of a sphere whose diameter is 9dm, if the circle is at a distance of 3dm from the center?

Ex. 909. The dihedral angles of a spherical pyramid are 40°, 80°, and 120°, and its edge is 9 ft. What is the volume of the pyramid ?

Ex. 910. The dihedrals of a trihedral angle whose vertex is at the center of a sphere are 75°, 90°, and 130°. What is the volume of the part of the sphere included by the faces of the trihedral, the radius of the sphere being 8m?

Ex. 911. What is the radius of a sphere which is equivalent to the sum of two spheres whose radii are respectively 4 in. and 7 in. ?

Ex. 912. How many cubic decimeters does a segment of a single base contain, if it is cut from a sphere 12dm in diameter, the altitude of the segment being 4dm?

Ex. 913. In a sphere whose diameter is 20 ft., what is the volume of a segment, the bases of which are on the same side of the center, one 3 ft. and the other 5 ft. from it?

Ex. 914. Find the area of the surface of a sphere inscribed in a cube whose surface is 726 sq. ft.

Ex. 915. A trirectangular triangle and a lune on the same sphere are in the ratio of 14:9. What is the angle of the lune, and what part of the surface of the sphere is the lune?

Ex. 916. Find the area of a spherical quadrilateral whose angles are 120°, 130°, 140°, and 150°, the volume of the sphere being 1000 cu. ft.

Ex. 917. The base of a cone of revolution is the great circle of a sphere, and its altitude is the radius of the sphere. What is the ratio of the surface of the sphere to the lateral surface of the cone ?

Ex. 918. The base of a cone is equal to a great circle of a sphere, and its altitude is equal to the diameter of the sphere. What is the ratio of their volumes ?

Ex. 919. Find the altitude of a zone whose area is equal to that of a great circle of the sphere, the radius of the sphere being 8dm.

Ex. 920. How many spherical bullets in. in diameter can be molded from a spherical piece of lead ft. in diameter ?

Ex. 921. A cannon ball put into a cylindrical tub 2 ft. in diameter causes the water in the tub to rise 2 in. What is the diameter of the cannon ball?

Ex. 922. A section parallel to the base of a hemisphere bisects its altitude. What is the ratio of the volumes of the spherical segments thus formed?

Ex. 923. The volume of a sphere is to that of the circumscribed cube as π is to 6.