Mensuration of Solids.

2. How many square feet are there in the convex surface of the frustum of a square pyramid, whose slant height is 10 feet, each side of the lower base 3 feet 4 inches, and each side of the upper base 2 feet 2 inches? Ans. 110.

3. What is the convex surface of the frustum of a heptagonal pyramid whose slant height is 55 feet, each side of the lower base 8 feet, and each side of the upper base 4 feet?

Ans. 2310 sq. ft.

PROBLEM V.

9. To find the solidity of a pyramid.

RULE.

Multiply the area of the base by the altitude and divide the product by three-the quotient will be the solidity.

QUEST.-9. How do you find the solidity of a pyramid ?

Mensuration of Solids.

2. Required the solidity of a square pyramid, each side of its base being 30 and its altitude 25.

Ans. 7500 solid feet.

3. How many solid yards are there in a triangular pyramid whose altitude is 90 feet, and each side of its base 3 yards?

Ans. 38,97117.

4. How many solid feet in a triangular pyramid the altitude of which is 14 feet 6 inches, and the three sides of its base 5, 6 and 7 feet?

Ans. 71,0352.

5. What is the solidity of a regular pentagonal pyramid, its altitude being 12 feet, and each side of its base 2 feet. Ans. 27,5276 solid feet.

6. How many solid feet in a regular hexagonal pyramid, whose altitude is 6,4 feet, and each side of the base 6 inches. Ans. 1,38564.

7. How many solid feet are contained in a hexagonal pyramid the height of which is 45 feet, and each side of the base 10 feet.

Ans. 3897,1143.

8. The spire of a church is an octagonal pyramid, each side of the base being 5 feet 10 inches, and its perpendicular height 45 feet. Within is a cavity, or hollow part, each side of the base of which is 4 feet 11 inches, and its perpendicular height 41 feet: how many yards of stone does the spire contain? Ans. 32,197353.

Mensuration of Solids.

PROBLEM VI.

10. To find the solidity of the frustum of a pyramid.

RULE.

Add together the areas of the two bases of the frustum and a geometrical mean proportional between them; and then multiply the sum by the altitude and take one-third of the product for the solidity.

EXAMPLES.

1. What is the solidity of the frustum of a pentagonal pyramid the area of the lower base being 16 and of the upper base 9 square feet,

the altitude being 7 feet.

First, 16 x 9=144: then √144=12 the mean.

Then, area of lower base = 16

QUEST.-10. How do you find the solidity of the frustum of a pyramid ?

Mensuration of Solids.

2. What is the number of solid feet in a piece of timber whose bases are squares, each side of the lower base being 15 inches, and each side of the upper base being 6 inches; the length being 24 feet?

Ans. 19,4776.

3. Required the solidity of a regular pentagonal frustum, whose altitude is 5 feet, each side of the lower base 18 inches, and each side of the upper base 6 inches.

Ans. 9,31925 solid feet.

4. What is the content of a regular hexagonal frustum, whose height is 6 feet, the side of the greater end 18 inches, and of the less end 12 inches?

Ans. 24,681724 cubic feet.

5. How many cubic feet in a square piece of timber, the areas of the two ends being 504 and 372 inches, and its length 31 feet?

Ans. 95,447.

6. What is the solidity of a squared piece of timber, its length being 18 feet, each side of the greater base 18 inches, and each side of the smaller 12 inches?

Ans. 28,5 cubic feet.

7. What is the solidity of the frustum of a regular hexagonal pyramid, the side of the greater end being 3 feet, that of the less 2 feet, and the height 12 feet?

Ans. 197,453776 solid feet.

Mensuration of the Round Bodies.

SECTION III.

OF THE MEASURES OF THE THREE ROUND BODIES.

1. To find the surface of a cylinder.

PROBLEM I.

RULE.

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.

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 (See I., Problem X.): we have 3,1416 × 12=37,6992-circumference of the base.

QUEST.-1. How do you find the surface of a cylinder?