Multiply the circumference by the height of the cylinder.

NOTE. The upright surface of any prism is found in the same manner. And the solidity of a cylinder is found as the prism in the last problem.

EXAMPLES.

1. What is the convex surface of a cylinder, whose length is 16 feet, and its diameter 2 feet 13 inches?

* DEMONSTRATION. If the periphery of the base be conceived to move, in a direction parallel to itself, it will generate the convex superficies of the cylinder; and therefore the said periphery, being multiplied by the length of the cylinder, will be equal to that superficies. Q. E. D.

NOTE. If twice the area of either of the ends be added to the convex surface, it will give the whole surface of the cylinder.

2. Required the convex surface of the length is 20 feet, and its diameter 2 feet.

cylinder, whose Ans. 125.664.

3. What is the convex surface of a cylinder, whose length is 18 feet 6 inches, and circumference 5 feet 4 inches? Ans. 98.

PROBLEM V.

To find the convex surface of a right cone.

RULE.

Multiply the circumference of the base by the slant height, or length of the side, and half the product will be the surface.

EXAMPLES.

1. If the diameter of the base be AB 5 feet, and the side of the cone AC 18; required the convex surface.

Ans. 90.

2. What is the convex surface of a cone, whose side is 20, and the circumference of its base 9 ?

3. Required

3. Required the convex surface of a cone, whose slant height is go feet, and the diameter of its base 8 feet 6 inches.

Ans, 667'59.

PROBLEM VI.

To find the convex surface of the frustum of a right cone.

RULE.

Multiply the sum of the perimeters of the two ends by the slant height or side of the frustum, and half the product will be the surface.

EXAMPLES.

1. If the circumferences of the two ends be 125 and 10°3, and the slant height AD 14; required the convex surface of the frustum ABCD.

2. What is the convex surface of the frustum of a cone, the slant height of the frustum being 125, and the circumferences of the two ends 6 and 84 ?

Ans. 90.

3. Required

3. Required the convex surface of the frustum of a cone, the side of the frustum being 10 feet 6 inches, and the circumferences of the two ends 2 feet 3 inches and 5 feet 4 inches. Ans. 391 3

PROBLEM VII.

To find the solidity of a cone, or any pyramid.

RULE.

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

EXAMPLES.

1. What is the solidity of a cone, whose height CD is 12 feet, and the diameter AB of the base 2?

2. What is the solid content of a pentagonal pyramid,

its height being 12 feet, and each side of its base 2 feet?

3. What is the content of a cone, its height being 10 feet, and the circumference of its base 9 feet?

Ans. 22*56693

4. Required the content of a triangular pyramid, its height being 14 feet 6 inches, and the 3 sides of its base 5, 6, 7. Ans. 710352.

5. What is the content of a hexagonal pyramid, whose height is 64, and each side of its base 6 inches?

Ans. 138564 feet.

PROBLEM VIII.

To find the solidity of the frustum of a cone or any pyramid.

RULES.

1. Add into one sum the areas of the two ends, and the mean proportional between them, or the square root of their product, and of that sum will be a mean area; and which, multiplied by the height of the frustum, will give the content.

VOL. II.

Ο

2. When