346. To find the contents of a prism,

Multiply the area of the base by the altitude, and the product will be the contents (Bk. VII. Prop. XIV).

EXAMPLES.

1. What are the contents of a square prism, each side of the square which forms the base being 16, and the altitude of the prism 30 feet?

ANALYSIS.-We first find the area of the square which forms the base, and then multiply by the altitude.

2. What are the contents of a cube, each side of which is 48 inches?

3. How many cubic feet in a block of marble, of which the length is 3 feet 2 inches, breadth 2 feet 8 inches, and height or thickness 5 feet?

4. How many gallons of water will a cistern contain, whose dimensions are the same as in the last example?

5. Required the solidity of a triangular prism, whose height is 20 feet, and area of the base 691.

347. A CYLINDER is a round body with circular ends. The line EF is called the axis or altitude, and the circular surface the convex surface of the cylinder.

348. To find the convex surface of a cylinder,

B

Multiply the circumference of the base by the altitude, and the product will be the convex surface (Bk. VIII. Prop. I).

347. What is a cylinder? What is the axis or altitude? What is the convex surface?

348. How do you find the convex surface?

EXAMPLES.

1. What is the convex surface of a cylinder, the diameter of whose base is 20 and the altitude 40 ?

2. What is the convex surface of a cylinder whose altitude is 28 feet and the circumference of its base 8 feet 4 inches?

3. What is the convex surface of a cylinder, the diameter of whose base is 15 inches and altitude 5 feet?

4. What is the convex surface of a cylinder, the diameter of whose base is 40 and altitude 50 feet?

349. To find the volume of a cylinder,

Multiply the area of the base by the altitude: the product will be the contents or volume (Bk. VIII. Prop. II).

EXAMPLES.

1. Required the contents of a cylinder of which the altitude is 11 feet, and the diameter of the base 16 feet.

ANALYSIS.-We first find the area of the base, and then multiply by the altitude: the product is the solidity.

2. What are the contents of a cylinder, the diameter of whose base is 40, and the altitude 29?

OPERATION.
162

= 256

.7854

area base, 201·0624

11

2111.6864

3. What are the contents of a cylinder, the diameter of whose base is 24, and the altitude 30?

4. What are the contents of a cylinder, the diameter of whose base is 32, and altitude 12?

5. What are the contents of a cylinder, the diameter of whose base is 25 feet, and altitude 15?

349. How do you find the contents of a cylinder?

350. A PYRAMID is a figure formed by several triangular planes united at the same point S, and terminating in the different sides of a plane figure, as ABCDE. The altitude of the pyramid is the line SO, drawn perpendicular to the base.

E

B

351. To find the contents of a pyramid.

Multiply the area of the base by the altitude, and divide the product by 3 (Bk. VII., Prop. XVII).

EXAMPLES.

1. Required the contents of a pyramid, the area of whose base is 86, and the altitude 24.

ANALYSIS.-We simply multiply the area of the base 86, by the altitude 24, and then divide the product by 3.

OPERATION.

86

24

3)2064

Ans. 688

2. What are the contents of a pyramid, the area of whose base is 365, and the altitude 36?

3. What are the contents of a pyramid, the area of whose base is 207, and altitude 36?

4. What are the contents of a pyramid, the area of whose base is 562, and altitude 30?

5. What are the contents of a pyramid, the area of whose base is 540, and altitude 32 ?

6. A pyramid has a rectangular base, the sides of which are 50 and 24; the altitude of the pyramid is 36: what are its contents?

7. A pyramid with a square base, of which each side is 15, has an altitude of 24: what are its contents?

350. What is a pyramid? What is the altitude of a pyramid ?

351. How do you find the contents of a pyramid ?

352. A CONE is a round body with a circular base, and tapering to a point called the vertex. The point C is the vertex, and the line CB is called the axis or altitude.

353. To find the contents of a cone.

B

Multiply the area of the base by the altitude, and divide the product by 3; or, multiply the area of the base by one-third of the altitude (Bk. VIII., Prop. V.)

EXAMPLES.

1. Required the contents of a cone, the diameter of whose base is 6, and the altitude 11.

ANALYSIS.-We first square the diameter, and multiply it by .7854, which gives 36 the area of the base. We next multiply

by the altitude, and then divide the product by 3.

OPERATION.
62 = 36

.7854

28.2744

11

3)311.0184

Ans. 103.6728

2. What are the contents of a cone, the diameter of whose base is 36, and the altitude 27 ?

3. What are the contents of a cone, the diameter of whose base is 35, and the altitude 27 ?

4. What are the contents of a cone, whose altitude is 27 feet, and the diameter of the base 20 feet?

GAUGING.

354. CASK-GUAGING is the method of finding the number of gallons which a cask contains, by measuring the external dimensions of the cask.

352. What is a cone?

What is the vertex? What is the axis?

353. How do you find the contents of a cone? 354. What is cask-gauging?

355. Casks are divided into four varieties, according to the curvature of their sides. To which of the varieties belongs, must be judged of by inspection.

any

cask

1. Of the least curvature.

2d Variety.

3d Variety.

4th Variety.

356. The first thing to be done is to find the mean diameter. To do this,

Divide the head diameter by the bung diameter, and find the quotient in the first column of the following table, marked Qu. Then if the bung diameter be multiplied by the number on the same line with it, and in the column answering to the proper variety, the product will be the true mean diameter, or the diame ter of a cylinder having the same altitude and the same contents with the cask proposed.

355. Into how many varieties are casks divided?

356. How do you find the mean diameter ?