138 PARALLELEPIPED, PRISM, CYLINDER.

XXIII. PARALLELEPIPED, PRISM, CYLINDER.

246. To find the volume of a parallelepiped, a prism, or a cylinder.

RULE. Multiply the area of the base by the height, and the product will be the volume.

(1) The area of the base of a parallelepiped is 5 square feet, and the height is 9 inches.

(2) The base of a prism is a triangle, the sides of which are 1 foot 1 inch, 1 foot 8 inches, and 1 foot 9 inches respectively; and the height of the prism is 1 foot 10 inches.

We must first find the area of the base by Art. 152.

1 foot 1 inch = 13 inches, 1 foot 8 inches=20 inches, 1 foot 9 inches=21 inches.

27 × 14 × 7×6=15876. The square root of 15876 is 126. Thus the area of the base is 126 square inches.

1 foot 10 inches = 22 inches.

126 × 22=2772. Thus the volume of the prism is 2772 cubic inches.

(3) The radius of the base of a cylinder is 5 inches, and the height is 16 inches.

The area of the base in square inches =78'54.

78 54 x 16=1256'64.

=

5 × 5 × 3 1416

Thus the volume is about 1256 64 cubic inches.

248. If we know the volume of a parallellepiped, a prism, or a cylinder, and also the area of the base, we can find the height by dividing the number which expresses the volume by the number which expresses the area of the base; and similarly if we know the volume and the height, we can find the area of the base.

249. Examples.

(1) The volume of a prism is a cubic foot, and the area of the base is 108 square inches; find the height.

1728

108

=16. Thus the height is 16 inches.

(2) The volume of a cylinder is 2000 cubic inches, and the height is 4 feet 2 inches: find the area of the base.

40. Thus the area of the base is 40 square inches.

250. It will be seen that the Rule in Art. 246 is the same as the second form of the Rule in Art. 239; and it will be instructive to attempt to explain the reason of this coincidence. We suppose that the beginner has convinced himself by the process of Art. 232, that the Rule holds for any rectangular parallelepiped; and we will try to shew that it will also hold for a right prism or a right circular cylinder.

251. Refer to the diagram of Art. 29. Let there be two right prisms with the same height, one having the triangle ABC for base, and the other having the rectangle ABDE for base. By the method of Art. 29 we can shew that the prism on the triangular base is half the prism on

140 PARALLELEPIPED, PRISM, CYLINDER.

the rectangular base. Hence it will follow that the Rule of Art. 246 holds for any right prism on a triangular base. Therefore the Rule will also hold for a right prism which has any rectilineal figure for its base; because such a base could be divided into triangles; and the prism could be divided into corresponding prisms with triangular bases, for each of which the Rule holds.

We are thus led to the notion that the volume of a right prism of given height depends only on the area of the base, and not on the shape of that base; and this may suggest that the Rule will also hold for a right circular cylinder. And we may infer that the Rule will also hold for other solids which are not called cylinders in ordinary language; for example, the shaft of a fluted column.

252. The Rule will also hold for oblique prisms and cylinders. The ground on which this rests is the following proposition: an oblique parallelepiped is equivalent to a rectangular parallelepiped which has the same base and an equal height. This proposition resembles that in Art. 28; and the mode of demonstration is similar; but instead of one adjustment of adding an area and subtracting an equal area, we shall here in general require two separate adjustments, of adding a volume and subtracting an equal volume.

253. We will now solve some exercises.

(1) A cubic inch of metal is to be drawn into a wire of an inch thick: find the length of the wire.

base

The wire will be

inches

=

a cylinder, having the radius of its of an inch. Thus the area of the base in square 007854. As the volume is 1 cubic inch,

3.1416

400

=

we divide 1 by 007854; and thus we obtain for the length of the wire 127.3 inches.

(2) Find the volume of a cylindrical shell, the height being 5 feet, the radius of the inner surface 3 inches, and the radius of the outer surface 4 inches.

By a cylindrical shell is meant the solid which remains when from a solid cylinder another cylinder with the same axis or with a parallel axis is removed; such bodies are usually called pipes or tubes.

By Art. 173, the area of the base in square inches =7 × 1 × 3·1416=219912; the height is 60 inches: therefore the volume in cubic inches 60 × 21.99121319472.

(3) The height of a cylinder is to be equal to the radius of the base, and the volume is to be 500 cubic inches: find the height.

Since the height is equal to the radius of the base, the product of 3.1416 into the cube of the number of inches in the radius must be equal to 500: hence the cube of the number of inches in the radius = =159 15. By ex

500 3.1416

tracting the cube root we obtain 5'419. Thus the radius is nearly 5:42 inches.

EXAMPLES. XXIII.

Find in cubic feet and inches the volumes of the prisms having the following dimensions:

1. Base 6 square feet 35 square inches; height 2 feet 6 inches.

2.

Base 15 square feet 135 square inches; height 3 feet 11 inches.

3. Base 23 square feet 115 square inches; height 4 feet 7 inches.

4. Base 35 square feet 123 square inches; height 5 feet 5 inches.

Find in cubic feet and inches the volumes of the triangular prisms having the following dimensions :

5. Sides of the base 7, 15, and 20 inches: height 45 inches.

6. Sides of the base 16, 25, and 39 inches: height 52 inches.

7. Sides of the base 13, 40, and 51 inches; height 58 inches.

8. Sides of the base 25, 33, and 52 inches; height 62 inches.

Find in cubic feet and decimals the volumes of the cylinders having the following dimensions:

9. Radius of base 2 feet; height 3 feet 6 inches.

10. Radius of base 2 feet 6 inches; height 4 feet 3 inches.

11. Radius of base 3 feet 6 inches; height 5 feet 9 inches.

12. Radius of base 5 feet 4 inches; height 6 feet 4 inches.

Find the heights of the prisms which have the following volumes and bases:

13. Volume 18 cubic feet 708 cubic inches; baso 6 square feet 100 square inches.

14. Volume 28 cubic feet 500 cubic inches; base 7 square feet 103 square inches.

15. Volume 36 cubic feet 349 cubic inches; base 9 square feet 35 square inches.

16. Volume 65 cubic feet 782 cubic inches; base 14 square feet 118 square inches.

Find the radii of the bases of the cylinders which have the following volumes and heights:

17. Volume 10000 cubic inches; height 4 feet 2 inches. 18. Volume 20 cubic feet; height 4 feet 71⁄2 inches. 19. Volume 50 cubic feet; height 5 feet 4 inches. 20. Volume 100 cubic feet; height 5 feet 10 inches.