Mensuration of the Round Bodies.

PROBLEM VI.

To find the solidity of the frustum of a cone.

RULE.

I. Add together the areas of the two ends and a geometrical mean between them.

II. Multiply this sum by one-third of the altitude and the product will be the solidity.

EXAMPLES.

1. How many cubic feet in the frustum of a cone whose altitude is 26 feet, and the diameters of the bases 22 and 18 feet?

First, 22x.7854-380.134-area of lower base:

and 18x.7854-254.47-area of upper base.

Then, 380.134 x 254.47 311.018 mean.

26

Then, (380.134+254,47+311.018)8195.39 which

is the solidity.

2. How many cubic feet in a piece of round timber the diameter of the greater end being 18 inches, and that of the less 9 inches, and the length 14.25 feet? Ans. 14.68943.

3. What is the solidity of a frustum, the altitude being 18, the diameter of the lower base 8, and of the upper 4? Ans. 527.7888.

4. If a cask, which is composed of two equal conic frustums joined together at their larger bases, have its bung diameter 28 inches, the head diameter 20 inches, and the length

Mensuration of the Round Bodies.

40 inches, how many gallons of wine will it contain, there being 231 cubic inches in a gallon?

PROBLEM VII.

Ans. 79.0613.

To find the surface of a sphere.

RULE.

Multiply the circumference of a great circle by the diameter, and the product will be the surface (Bk. VI. Th. xxiii).

2. What is the surface of a sphere whose diameter is 24?

Ans. 1809.5616.

3. Required the surface of a sphere whose diameter is 7921 miles. Ans. 197111024 sq. miles.

4. What is the surface of a sphere the circumference of whose great circle is 78.54?

Ans. 1963.5.

5. What is the surface of a sphere whose diameter is 1 Ans. 5.58506 sq. ft.

feet?

PROBLEM VIII.

To find the convex surface of a spherical zone.

RULE.

Multiply the height of the zone by the circumference of a great circle of the sphere, and the product will be the convex surface (Bk. VI. Th. xxiv).

2. The diameter of a sphere is 12 feet: what will be the surface of a zone whose altitude is 2 feet?

Ans. 78.54 sq. ft.

3. The diameter of a sphere is 21 inches: what is the surface of a zone whose height is 4 inches?

Ans. 296.8812 sq. in.

4. The diameter of a sphere is 25 feet and the height of the zone 4 feet: what is the surface of the zone?

Ans. 314.16 sq. ft.

5. The diameter of a sphere is 9, and the height of a zone 3 feet: what is the surface of the zone?

Ans. 84.8232.

PROBLEM IX.

To find the solidity of a sphere.

RULE I.

Multiply the surface by one-third of the radius and the product will be the solidity (Bk. VI. Th. xxv)'.

2. The diameter of a sphere is 7957.8: what is its solidity?

Ans. 263863122758.4778.

3. The diameter of a sphere is 24 yards: what is its solid contents? Ans. 7238.2464 cubic yds.

4. The diameter of a sphere is 8: what is its solidity?

Ans. 268.0832.

5. The diameter of a sphere is 16: what is its solidity? Ans. 2144.6656.

RULE II.

Cube the diameter and multiply the number thus found, by the decimal .5236, and the product will be the solidity.

EXAMPLES.

1. What is the solidity of a sphere whose diameter is 20?

Ans. 4188.8.

2. What is the solidity of a sphere whose diameter is 6? Ans. 113.0976.

3. What is the solidity of a sphere whose diameter is 10?

22*

Ans. 523.6.

Mensuration of the Round Bodies.

PROBLEM X.

To find the solidity of a spherical segment with one base.

RULE.

I. To three times the square of the radius of the base, add the square of the height.

II. Multiply this sum by the height, and the product by the decimal .5236, the result will be the solidity of the segment.

Then, 163 X4 X.5236-341.3872 solid feet, which is the solidity of the segment.

2. What is the solidity of the segment of a sphere whose height is 4, and the radius of its base 8? Ans. 435.6352.

3. What is the solidity of a spherical segment, the diameter of its base being 17.23368, and its height 4.5?

Ans. 572.5566.

4. What is the solidity of a spherical segment, the diameter of the sphere being 8, and the height of the segment 2 feet? Ans. 41.888 cubic ft.

5. What is the solidity of a segment, when the diameter of the sphere is 20, and the altitude of the segment 9 feet? Ans. 1781.2872 cubic ft.