2. Spherical segment not including the center.

Given Spherical segment generated by figure ACDB; semicircle XABY; AC=r; BD=r'; radius of sphere = R; altitude =

CD=h

Required: To find the volume of the spherical segment.

Computation: The ▲ ACO and BDO generate cones of revolution (?) (671).

The volume of spherical segment ACDB

B

X

O

= volume of spherical sector ABO plus the volume of cone ACO

minus the volume of cone BDO. Each of these volumes can be determined from formulas already established.

3. Spherical segment including the center.

Given Spherical segment generated by figure BDSR; etc.

:

Required (?). Computation: The same as that of case 2, except that both cones, BDO and RSO, are added.

1. Prove that the area of the surface of a sphere is equal to the square of the diameter multiplied by π, that is, SD2.

2. Prove that the volume of a sphere is equal to one sixth the cube of the diameter multiplied by π, that is, V = πD3.

3. The surface of a sphere is equal to the cylindrical surface of the circumscribed cylinder.

4. The total surface of a hemisphere is three fourths the surface of the sphere.

5. The volume of a sphere is two thirds the volume of the circumscribed cylinder.

6. Upon the same circle as a base are constructed a hemisphere, a cylinder of revolution, and a cone of revolution, all having the same altitude. Prove that their total areas are 3πR2, 4πR2, πR2(1 + √2), respectively, and their volumes are } πR3, πR3, } #R3, respectively.

7. Two zones on the same sphere, or on equal spheres, are to each other as their altitudes.

8. The area of the surface of a sphere is equal to the area of the circle whose radius is the diameter of the sphere.

9. Show that the formula for the volume of a spherical segment of one base reduces to the correct formula for the volume of a hemisphere, when the base of the segment is a great circle; and to the correct formula for the volume of a sphere when the planes are both tangent.

10. In an equilateral triangle is inscribed a circle, and the figure is revolved about an altitude of the triangle as an axis. Prove,

(a) That the surface generated by the circumference is two thirds the lateral surface generated by the triangle.

(b) That the volume generated by the circle is four ninths the volume generated by the triangle.

2r√3

r√3

11. Derive a formula for the surface of a sphere, containing only V and π.

12. Derive a formula for the volume of a sphere, containing only S and π.

13. In a circle whose radius is R, there are inscribed a square and an equilateral triangle having their bases parallel; the whole figure is then revolved about the diameter perpendicular to the base of the triangle. Find, in terms of R,

(a) The total areas of the three surfaces generated; (b) The volumes of the three solids generated.

14. If a cylinder of revolution having its altitude equal to the diameter of its base, and a cone of revolution having its slant height equal to the diameter of its base are both inscribed in a sphere,

(a) The total area of the cylinder is a mean proportional between the area of the surface of the sphere and the total area of the cone;

(b) The volume of the cylinder is a mean proportional between the volume of the sphere and the volume of the cone.

15. About a circle whose radius is R, there are circumscribed a square and an equilateral triangle having their bases in the same straight line. The whole figure is then revolved about an altitude of the triangle. Find, in terms of R,

(a) The total areas of the three surfaces generated.

(b) The volumes of the three surfaces generated.

16. If a cylinder of revolution having its altitude equal to the diameter of its base, and a cone of revolution having its slant height equal to the diameter of its base, be circumscribed about a sphere,

(a) The total area of the cylinder is a mean proportional between the area of the surface of the sphere and the total area of the cone;

(b) The volume of the cylinder is a mean proportional between the volume of the sphere and the volume of the cone.

17. The line joining the centers of two intersecting spherical surfaces is perpendicular to the plane of the intersection at the center of the intersection.

18. A cube and a sphere have equal surfaces; show that the sphere has the greater volume.

19. Prove that the parallel of latitude through a point having 30° north latitude bisects the surface of the northern hemisphere.

20. Prove that in order that the eye may observe one sixth of the surface of a sphere, it must be at a distance from the center of the sphere equal to of the radius.

Proof: Zone TT

.. AB = | diam. = R.

surface of sphere (Hyp.).

Hence, BC = } R.

E

C

In rt. ▲ ETC, TC2 = EC.BC (?) ; .. R2 = EC. R, or EC

(Explain.)

= R

Q.E.D.

21. How many miles above the surface of the earth (diameter of earth = 7960 mi.) must a person be in order that he may see one sixth of the earth's surface?

22. If the area of a zone of one base is a mean proportional between the area of the remaining zone of the sphere and the area of the entire sphere, the altitude of the zone is R( √5 − 1).

23. The area of a lune is to the area of a trirectangular spherical triangle as the angle of the lune is to 45°.

24. A cone, a sphere, and a cylinder have the same diameters and altitudes. Prove that their volumes are in arithmetical progression.

25. The surface of a sphere bears the same ratio to the total surface of the circumscribed cylinder of revolution, as the volume of the sphere bears to the volume of the cylinder.

26. The smallest circle upon a sphere, whose plane passes through a given point within the sphere, is the circle whose plane is perpendicular to the diameter through the given point.

27. What part of the surface of the earth could one see if he were at the distance of a diameter above the surface?

28. Prove that if any number of lines in space be drawn through a point, and from any other point perpendiculars to these lines be drawn, the feet of all of these perpendiculars lie on the surface of a sphere.

29. The volume of a sphere is to the volume of the circumscribed cube as π: 6. The volume of a sphere is to the volume of the inscribed cube as π: √3.

30. There are five spheres that touch the four planes of the faces of a tetrahedron.

31. If two angles of a spherical triangle are supplementary, the sides of the polar triangle, opposite these angles, are supplementary.

32. A square, whose side is a, is revolved about a diagonal, and also about an axis bisecting two opposite sides. Which of these figures contains the greater volume? Which has the greater surface?

33. Find the area of the surface, and the volume of a sphere whose radius is 6.

34. Find the area of a zone whose altitude is 4 on a sphere whose radius is 14.

35. Find the area of a lune whose angle is 30° on a sphere whose radius is 8 in.

36. Find the area of a spherical triangle whose angles are 110°, 41°, 92°, on a sphere whose radius is 10.

37. Find the volume of a sphere whose radius is 5.

38. Find the volume of a spherical pyramid whose base is 35 sq. in., on a sphere whose radius is 12 in.

39. Find the area of a spherical polygon whose angles are 87°, 108°, 121°, 128°, on a sphere whose radius is 25.

40. What is the radius of a sphere whose surface is 1386 sq. yd.?

41. What is the radius of a sphere whose volume is

42. What is the area of the surface of a sphere whose volume is 288 cu. ft.?

43. What is the volume of a sphere, the area of whose surface is 2464 sq. in.?

44. Find the area of a zone whose altitude is 31, if the radius of the sphere is 71.

45. Find the volume of a spherical sector the altitude of whose base is 5 in. if the radius of the sphere is 6 in.

46. Find the diameter, the circumference of a great circle, and the volume of a sphere the area of whose surface is 25 π sq. ft.

47. By how many cubic inches is a 9-in. cube greater than a 9-in. sphere?

48. The radius of a sphere is 15, and the angles of the base of a spherical pyramid are 160°, 127°, 96°, 145°, and 117°. Find the volume of the pyramid.

49. A cylindrical vessel 10 in. in diameter contains a liquid. A metal ball is immersed in the liquid and the surface rises in. What is the diameter of the ball?

50. If a sphere 3 ft. in diameter weighs 99 lbs., what will a sphere of the same material 4 ft. in diameter weigh?

51. The radii of the bases of a frustum of a cone of revolution are 5 in. and 6 in., and the altitude of the frustum is 19 in. What is the diameter of an equivalent sphere?

52. What is the radius of a sphere whose surface is equivalent to the total surface of a right circular cylinder having an altitude equal to 21, and radius of the base equal to 6?

53. Find the volume generated by the revolution of an equilateral triangle inscribed in a circle whose radius is 8, about an altitude of the triangle as an axis. (See Fig. of Ex. 55.)

54. In the figure of No. 55, find the volume of the segment generated by the figure AED revolving about CD as an axis.

55. Find the area of the surface, and the volume of the sphere generated by a circle that is circumscribed about an equilateral triangle whose side is 10.

56. Circumscribing a sphere whose radius is 18, is a cylinder of revolution. Compare their total areas. Their volumes.

ROBBINS' SOLID GEOM. -26