Multiply the circumference of the base by the altitude, and the product will be the convex surface; and to this, add the areas of the two bases, when the entire surface is required (Bk. VI. Th. ii).

EXAMPLES.

1. What is the entire surface of the cylinder in which AB, the diameter of the base, is 12 feet, and the altitude EF 30 feet?

First, to find the circumference of the base, (Prob. X. page 180): we have

3.1416 × 12=37.6992-circumference of

the base.

Then, 37.6992 x 30=1130.9760-convex surface.

Also, 12=144

base.

Then,

B

2. What is the convex surface of a cylinder, the diameter

of whose base i 20, and the altitude 50 feet?

Mensuration of the Round Bodies.

3. Required the entire surface of a cylinder, whose altitude is 20 feet, and the diameter of the base 2 feet.

Ans. 131.9472 ft.

4. What is the convex surface of a cylinder, the diameter of whose base is 30 inches, and altitude 5 feet?

Ans. 5654.88 sq. in.

5. Required the convex surface of a cylinder, whose alti tude is 14 feet, and the circumference of the base 8 feet 4 inches. Ans. 116.6666, &c., sq. ft.

PROBLEM II.

To find the solidity of a cylinder.

RULE.

Multiply the area of the base by the altitude, and the product will be the solidity.

EXAMPLES.

1. What is the solidity of a cylinder,

the diameter of whose base is 40 feet, and altitude EF, 25 feet?

First, to find the area of the base, we have (Prob. xv. page 231).

4021600: then, 1600×.7854-1256.64.

area of the base.

Then, 1256.64 × 25-31416 solid feet, which is the solidity.

2. What is the solidity of a cylinder, the diameter of whose base is 30 feet, and altitude 50 feet?

Ans. 35343 cubic ft.

Mensuration of the Round Bodies.

3. What is the solidity of a cylinder whose height is 5 feet, and the diameter of the end 2 feet?

Ans. 15.708 solid ft.

4. What is the solidity of a cylinder whose height is 20 feet, and the circumference of the base 20 feet?

Ans. 636.64 cubic ft.

5. The circumference of the base of a cylinder is 20 feet, and the altitude 19.318 feet: what is the solidity?

Ans. 614.93 cubic ft.

6. What is the solidity of a cylinder whose altitude is 12 feet, and the diameter of its base 15 feet?

Ans. 2120.58 cubic ft.

7. Required the solidity of a cylinder whose altitude is 20 feet, and the circumference of whose base is 5 feet 6 inches? Ans. 48.1459 cubic ft.

8. What is the solidity of a cylinder, the circumference of whose base is 38 feet, and altitude 25 feet?

Ans. 2872.838 cubic ft.

9. What is the solidity of a cylinder, the circumference of whose base is 40 feet, and altitude 30 feet?

10. The diameter of the base of a cylinder is 84 yards, and the altitude 21 feet: how many solid or cubic yards does it contain ? Ans. 38792.4768.

PROBLEM III.

To find the surface of a cone.

RULE.

Multiply the circumference of the base by the slant height, and divide the product by 2; the quotient will be the convex surface, to which add the area of the base, when the entire surface is equired (Bk. VI. Th. viii).

Mensuration of the Round Bodies.

EXAMPLES.

1. What is the convex surface of the cone whose vertex is C, the diameter AD, of its base being 81 feet, and the side CA, 50 feet.

B

First, 3.1416x8=26.7036=circumference of base.

2. Required the entire surface of a cone whose side is 36 and the diameter of its base 18 feet.

Ans. 1272.348 sq. ft.

3. The diameter of the base is 3 feet, and the slant height what is the convex surface of the cone?

15 feet

Ans. 70.686 sq. ft.

4. The diameter of the base of a cone is 4,5 feet, and tha slant height 20 feet: what is the entire surface?

Ans. 157.27635 sq. ft.

5. The circumference of the base of a cone is 10.75, and the slant height is 18.25: what is the entire surface?

PROBLEM IV.

Ans. 107.29021 sq. ft.

To find the solidity of a cone.

RULE.

Multiply the area of the base by the altitude; and divide the pro duct by 3, the quotient will be the solidity (Bk. VI. Th. xviii).

Mensuration of the Round Bodies.

EXAMPLES.

1. What is the solidity of a cone, the area of whose base is 380 square feet, and altitude CB, 48 feet?

2. Required the solidity of a cone whose altitude is 27 teet, and the diameter of the base 10 feet.

3. Required the solidity of a

Ans. 706.86 cubic ft.

cone whose altitude is 10

feet, and the circumference of its base 9 feet?

Ans. 22.5609 cubic ft.

4. What is the solidity of a cone, the diameter of whose base is 18 inches, and altitude 15 feet?

Ans. 8.83575 cubic ft.

5. The circumference of the base of a cone is 40 feet, and

the altitude 50 feet: what is the solidity?

Ans. 2122.1333 solid ft.