When the ends are regular plane figures; the mean area will be found by multiplying of the corresponding tabular number belonging to the polygon, either by the sum arising by adding together the square of a side of each end and the product of the two sides, or by the quotient of the difference of their cubes divided by their difference, or by the sum arising from the square of their half difference added to 3 times the square of their half sum.

3. And in the frustum of a cone, the mean area is found by multiplying 2618, or of 7854, either by the sum arising by adding together the squares of the two diameters and the product of the two, or by the difference of their cubes divided by their difference, or by the square of half their difference added to 3 times the square of their half sum.

Or, if the circumferences be used in like manner, instead of their diameters, the multiplier will be 02654. 2652.6

EXAMPLES.

1. What is the content of the frustum of a cone, whose height is 20 inches, and the diameters of its two ends 28 and 20 inches?

2. Required the content of a pentagonal frustum, whose height is 5 feet, each side of the base 1 foot 6 inches, and each side of the less end 6 inches.

3. What is the solidity of the frustum of a cone, the altitude being 25, the circumference at the greater end being 20, and at the less end 10?

Ans. 464 205.

4. How many solid feet are in a piece of timber, whose bases are squares, each side of the greater end being 15 inches, and each side of the less end 6 inches; also the length, or perpendicular altitude, is 24 feet?

Ans. 19

5. To find the content of the frustum of a cone, the altitude being 18, the greatest diameter 8, and the least 4.

Ans. 527-7888.

6. What

6. What is the solidity of a hexagonal frustum, the height being 6 feet, the side of the greater end 18 inches, and that of the less 12 inches ?.. Ans. 24681722.

PROBLEM IX.

To find the solidity of a wedge.

RULE.

To the length of the edge add twice the length of the back or base, and reserve the sum; multiply the height of the wedge by the breadth of the base; then multiply this product by the reserved sum, and of the last product will be the content.

EXAMPLES.

1. What is the content, in feet, of a wedge, whose altitude EP is 14 inches, its edge EF 21 inches, the length of its base BC 32 inches, and its breadth AB 4 inches?

2. Required the content of a wedge, the length and breadth of the base being 70 and 30 inches, the length of the edge 110 inches, and the height 34'29016.

Ans. 24-8048.

PROBLEM X.

To find the solidity of a prismoid..

NOTE. A prismoid differs only from the frustum of a pyramid in not having its opposite ends similar planes.

RULE.

Add into one sum the areas of the two ends and 4 times the middle section parallel to them, and of that sum will be a mean area; which, being multiplied by the height, will give the content.

NOTE. The length of the middle section is equal to half the sum of the lengths of the two ends; and its breadth is equal to half the sum of the breadths of the two ends.

EXAMPLES.

1. How many cubic feet are there in a stone, whose ends are rectangles, the length and breadth of one being 14 and 12 inches; the corresponding sides of the other 6 and 4 inches; and the perpendicular height 30 feet?

14 12

168

B