792. Cor. 4. The volumes of spherical sectors of the same sphere, or equal spheres, are to each other as the zones which form their bases, or as the altitudes of these zones. For, let V and V denote the volumes of two spherical sectors, 2 and 2 the zones which form their bases, H and H' the altitudes of these zones, and R the radius of the sphere. 793. Cor. 5. The volume of a spherical segment of one base, less than a hemisphere, generated by the revolution of a circular sector A O B about the diameter A D, may be found by subtracting the volume of the cone of revolution generated by OBC from that of the spherical sector A O B. In like manner, the volume of a spherical segment of one base, greater than a hemisphere, generated by the revolution of A B'C' may be found by adding the volume of the cone of revolution generated by 0 B'C' to that of the spherical sector generated by A O B'. 794. CoR. 6. The volume of a spherical segment of two bases, generated by the revolution of C B B'C' about the diameter A D, may be found by subtracting the volume of the segment of one base generated by A B C from that of the segment of one base generated by A B'C'. EXERCISES. 1. Given a sphere whose diameter is 20 inches ; find the circumference of a small circle whose plane cuts the diameter 4 inches from the centre. 2. Construct, on the spherical blackboard, spherical angles of 30°, 45°, 90°, 120°, 150° and 135o. 3. Construct, on the spherical blackboard, a spherical triangle, whose sides are 100°, 80° and 70° respectively. What is true of its polar triangle ? 4. Find the surface and volume of a sphere whose radius is 10 inches ; also find the area of a spherical triangle on this sphere, the angles of the triangle being 80°, 85° and 100° respectively. 5. If 7 equidistant planes cut a sphere, each perpendicular to the same diameter, what are the relative areas of the zones? 6. Given, two mutually equiangular triangles on spheres whose radii are 10 inches and 40 inches respectively; what are their relative areas ? 7. Let V denote the volume of a spherical pyramid, S its base, E the spherical excess of its base, and R the radius of the sphere; show that S= } * R? E, and V = 5 a R$ E. 8. Given, the volume of a sphere 1728 inches ; find its radius. 9. Find the ratio of the surfaces, and the ratio of the volumes, of a cube and of the inscribed sphere. 10. Find the ratio of the surfaces, and the ratio of the volumes, of a sphere and the circumscribed cylinder. 11. Let V denote the volume and H the altitude of the spherical segment of one base, and R the radius of the sphere; show that V =q Ho (R – }H). Also, find V when R=12 and H= 3. 12. Given, a sphere 2 feet in diameter; find the volume of a segment of the sphere included between two parallel planes, one at 3 and the other at 9 inches from the centre. (Two solutions.) 13. A sphere 4 inches in diameter is bored through the centre with a two-inch auger; find the volume remaining. THE END. University Press : John Wilson & Son, Cambridge. |