Proposition 12. Theorem.

802. The volume of a spherical segment is equal to half the sum of its bases multiplied by its altitude, plus the volume of a sphere of which this altitude is the diameter.

Hyp. Let AB be the arc of a O, and CD the projection of the chord AB on the diameter OM.

Let AC r', BD, CD =1, OA=R, and denote the volume of the segment generated by revolving the circular segment AHBDC about OM by V.

To prove Vi̟πh (12 + r22) +‡πh3.
Proof. Draw the radii OA, OB.

B

H

M

C

The volume generated by AHBDC is the sum of the spherical sector generated by OAHB and the cone generated by OAC, diminished by the cone generated by OBD. Vol. sph. sect. OAHB7R.

Vol. of cone OAC "OC.

Vol. of cone

... V =

R2 –

(800)

(778)

But OC OD, R2 2 OC', and R' - 72 = OD'.

— r22 =

= }πh[3R2 — (OC2 + OC. OD + OD2)]

=

OC OD 1,
-

20C. OD OD' = h2, and

Since

... OC

OC. OD + OD2 = {(OC' + OD′) — 2*

2

803. COR. 1. If the segment has but one base, as the volume generated by MABD, the radius = 0, and we have

Therefore, the volume of a spherical segment of one base is equal to half the cylinder having the same base and the same altitude, plus the sphere of which this altitude is the diameter.

804. COR. 2. When the segment has but one base,

805. Cor. 3. Let V' denote the volume of a frustum of a cone generated by the trapezoid ABDC about OM, and v the volume generated by the circular segment AHB.

Then

V' = }πh(r2 + p22 +rp'),

and

V = {πh(r2 + r'2) + fπh3.

Subtracting, v = {πh(r2 + r22 + h2 — 2rr′)

(782)
(802)

Therefore, the volume generated by a circular segment revolving about a diameter exterior to its surface, is equal to one-sixth of the cylinder whose radius is the chord of the segment and whose altitude is the projection of this chord on the axis.

EXERCISES.

THEOREMS.

1. The lateral area of a cylinder of revolution is equal to the area of a circle whose radius is a mean proportional between the altitude of the cylinder and the diameter of its base.

2. If the slant height of a cone of revolution is equal to the diameter of its base, its lateral area is double the area of its base.

3. The volume of a cylinder of revolution is equal to the product of its lateral area by half its radius.

4. A plane through two elements of a cylinder of revolution cuts the base in a chord which subtends at its centre

π

an angle of compare the lateral areas of the two parts

:

of the cylinder.

5. A rectangle revolves successively about two adjacent sides whose lengths are a and b compare the volumes of the two cylinders that are generated.

6. The two legs of a right triangle are a and b : find the area of the surface generated when the triangle revolves about its hypotenuse.

7. Prove that a sphere may be inscribed in a cylinder of revolution, and that it will touch it along the circumference of a great circle.

8. The lateral area of a given cone of revolution is double the area of its base: find the ratio of its altitude to the radius of its base.

9. On each base of a frustum of a cone of revolution, a cone stands having its vertex in the centre of the other base find the radius of the circle of intersection of the two cones, the radii of the bases being r, and r

10. If the altitude of a cylinder of revolution be equal to the diameter of its base,* the volume is equal to the product of its total area by one-third of its radius.

11. If the slant height of a cone of revolution be equal to the diameter of its base, its total area is to the area of the inscribed sphere as 9: 4.

12. In a frustum of a cone of revolution the inclination of the slant height to one base is 45°: find the lateral area, the radii of the bases being r and 12.

13. If the radius of a sphere is bisected at right angles by a plane, the two zones into which the surface of the sphere is divided are to each other as 3 : 1.

14. If a cylinder and cone, each equilateral, be inscribed in a sphere, the total area of the cylinder is a mean proportional between the total area of the cone and the area of the sphere. The same is true of the volumes of these

bodies.

15. If a cylinder and cone, each equilateral, be circumscribed about a sphere, the total area of the cylinder is a mean proportional between the total area of the cone and the area of the sphere. The same is true of the volumes.

16. A cone of revolution whose vertical angle is 60°, is circumscribed about a sphere: compare the area of the sphere and the lateral area of the cone. Compare their volumes.

17. The base of a cone is equal to a great circle of a sphere, and the altitude of the cone is equal to a diameter of the sphere: compare the volumes of the cone and sphere.

18. The volume of a cone of revolution is equal to the area of its generating rectangle multiplied by the circumference generated by the point of intersection of the diagonals of the rectangle.

19. The volume of a sphere is to the volume of the circumscribed cube as π: 6.

*Called an equilateral cylinder.

+ An equilateral cone,

20. The volume of a sphere is to the volume of the inscribed cube as π: 2.

21. If d is the distance of a point P from the centre of a sphere whose radius is R, the sum of the squares of the six segments of three chords at right angles to each other passing through P is 6R-2d3.

22. If h is the height of an aeronaut, and

of the earth, the extent of surface visible =

NUMERICAL EXERCISES.

R is the radius

2πR'h
R+h

23. Find the lateral area, total area, and volume, of a cylinder of revolution, the radius of the base being 4 and the altitude 10.

24. Find the lateral area, total area, and volume, of a cone of revolution, the radius of the base being 4 and the altitude 10.

25. Find the lateral area, total area, and volume, of a frustum of a cone of revolution, the radii of the bases being 7 and 2 and the altitude 3.

26. Find the lateral area of a frustum of a cone of revolution, the radii of the bases being 21 and 6 inches and the altitude 36 inches. Ans. 3308.1 cu. ins. 27. Find the volume of a frustum of a cone of revolution, the radii of the bases being 4 and 2 feet and the altitude 9 feet. Ans. 65.97 cu. ft.

28. The slant height of a cone of revolution is 4 feet: how far from the vertex must the slant height be cut by a plane parallel to the base that the lateral area may be divided into two equivalent parts?

29. The altitude of a cone of revolution is equal to the diameter of its base: find the ratio of the area of the base to the lateral area.

30. Find the volume of an equilateral cylinder in terms of its total area.

See Ex. (10).