NOTE. Multiply the circumference and diameter together and divide the product by; the quotient is the area of one end.

Examples.

1. What is the solidity of a cylinder that has a diameter 9 feet, and is 21 feet long?

9.5 diam. X 29.85+ circumference 283.575170.88f area 21 length 1488 69 solidity Ans.

2. What is the solidity of a cylinder, whose diameter is 7 inches, and circumference 22, and is 20 feet long?

7x22-154-1-38.5 area x20=770-1445,50

CASE V.

[Ans.

To measure, or find the solidity of a cone or pyramid. NOTE. A Cone is a figure standing upon a base, and of a true slant or taper from the largest end, to a point or vertex. Sometimes the base is circular, square, triangular, or in form of a parallelogram.

44

Figure 1, represents a circular cone; figure 2, a a triangular cone; figure 3, a square cone; and figure 4, a parallelogramical cone.

RULE. Find the superficial area of the base, and multiply it into one third of the perpendicular height of the cone, and the product will be the solidity.

Examples.

1. What is the solidity of figure first; perpendicular 21 feet, and the diameter of the circle at the base 9 feet?

9.5 diameter, 29.85 circumference.

Half diam. and half circum. 4·75 × 14·925=70·89† area of base. Area 70-89 X 21-496-23+ feet, solidity.

2. What is the solidity of figure 2d; perpendicular 21 feet, sides of its triangle 10, and perpendicular of its triangle 81 ?

10X8185÷÷1=42•5×21÷297·5

solidity.

3. What is the solidity of figure third; perpendicular height 21, and the sides of its base 9 feet?

9X9 81X21-567 ft. Ans.

4. What is the solidity of figure fourth: perpendicular height equal 21, and sides 15 by 91?

Ans. 997:5 ft.

CASE VI.

To find the solidity of the frustum of a cone, &c.* RULE I. Find the superficial area of both ends, add the two areas together, and reserve the sum; multiply the two areas together, extract the square

*A frustum of a cone is a piece cut off from the cone parallel to the base; see the figures in case 5th, A B C D are frustums.

P

root of the product, and add the root to the reserved sum, and multiply the sum by one third the perpendicular height, the product is the solidity required.

RULE II. If the cone is exactly square at both ends; multiply a side of the greatest, by a side of the least square; also find the difference between the two sides; square the difference; add one third of the square to the product of the two sides; multiply the last sum by the length, the product is the solidity.

Examples.

1. What is the solidity of the frustum A B C D. figure first, case 5th; largest diameter 9; smallest 3; and length 14?

Largest diam. 9.5 X 9.5 X 7854-70.88† area largest end. Smallest diam. 3·0 × 3·0 ×·7854 7·06† area least end.

2

Arcas70-88 X 7·06= √ 500·4128=22:36 root added.

2. What is the solidity of the frustum A B C D figure second, case fifth; the sides of its largest triangle 10, and its perpendicular 8; sides of the least triangle 3.5 and the perpendicular of its triangle 3; and length 14?

2

10X 442-5 area largest end. 3.5 X 1.5 5.25 area small. end.

Areas 42.5X5•25= √ 223·1250=14·93 root added

62.68X42=292.5 solidity [Ans.

3. What is the solidity of the frustum A B C D figure third, case fifth; sides of its largest square 9, and the smallest 3, and length 14?

By rule second.

Sides of the two squares 9x3=27 pro.

9—3—6 dif.×6=36÷3

= 12 add.

39×14=546 solidity.

[Ans.

4. What is the solidity of the frustum A B C D figure fourth, case fifth; sides of the largest end 15 by 9; sides of the smallest end 5 by 3.2; and 14 in length?

Sides of the largest end 15 X 9.5=142·5 area.
Sides of the smallest end 5×3·2= 160 area.

2

Areas 142.5X16= √ 2280·0=47•7† root 47-7 added.

5. What is the solidity of the that is 10 feet square at one end, and is 32 feet long?

206.2 × 14÷1 equal to 962-2† Ans. frustum of a cone and 4 at the other, Ans. 1664 ft.

6. What is the solidity of the frustum of a cone that is 16 inches square at one end, and 12 at the other, and is 16 feet long?

Ans. 21.

7. What is the solidity of the frustum of a cone that is 24 inches square at one end and 21 at the other, and is 10 feet long? Ans. 35

CASE VII.

To find the solidity of the segment of a cone, as the parts above A B in the figures, case 5th.

This part forms a cone of itself, and must be measured by case fifth.

CASE VIII.

To find the solidity of a wedge, when the edge and large end are of equal width.

RULE. Multiply the length and width together and that product by half the thickness, the last product is the solidity.

Examples.

1. What is the solidity of a wedge that is 12 feet long, 12 wide and 6 thick, at the large end?

12X12=144×6=432 solidity. Ans.

2. What is the solidity of a wedge that is 10 feet long, 6 inches wide and 6 inches thick?

Ans. 1 ft.

CASE IX.

To find the solidity of a wedge when the edge is narrower than the large end.

RULE. To the width of the edge add twice the width of the large end, and reserve the sum; multiply the length of the wedge by the thickness of the large end; multiply this last product and the reserved sum together, and divide by 6; the quotient is the solidity.

Examples.

1. What is the solidity of a wedge that is 12 feet long, and 18 inches wide at the largest end, and 12 inches wide at the edge; and six inches thick at the large end?

864 product.

Pro. 864 X res. sum 48-4147266912 in. or 4 ft. Ans.

2. What is the solidity of a wedge that is 20 long, 15 wide at the large end, and 10 at the edge; and 12 thick?

Ans. 1600.