1. What is the solid content of a pentagonal pyramid, the side of its base being 2 feet, and perpendicular height 20 feet?

Here (1.7204774 X 4 X 20) ÷ 3 = 45.87939 feet.

2. What is the solid content of a triangular pyramid, each side of its base being 4 feet, and perpendicular height 20 feet? Ans. 46.188 feet.

3. What is the superficial and solid content of a square pyramid, each side of its base being 8 feet, and slant height 20 feet? Ans. sup. 384ft. sol. 418.0458ft.

4. What is the solid content of an octagonal pyramid, the side of its base being 3 feet, and perpendicular height 24 feet? Ans. 347.6467. 5. What is the solid content of a cone, the diameter of its base being 2 feet 6 inches, and perpendicular altitude 12 feet? Ans. 19.635. 6. What is the superficial and solid content of a cone, its slant height being 18 feet, and the diameter of its base 8 feet 6 inches?

Ans. sup. 297.0775ft. sol. 330.8276ft. 7. What is the curve surface and solidity of a cone, its slant height being 25 feet, and the circumference of its base 6 feet 9 inches?

Ans. sur. 84.375ft. sol. 30.186ft.

PROBLEM 4.-To find the solidity of a frustum of a pyramid

and cone.

RULE. Add together the areas of the two ends and a mean proportional between them (that is the square root of their product) multiply this sum by the perpendicular height, and of the product will be the solidity.

EXAMPLES.

1. What is the solid content of a frustum of a cone, its height being 15 feet, the greater diameter 5 feet, and less 4 feet?

52 = 25

42 16

5 X 4 =20

61 X .7854 X 5 239.547 feet. Ans.

2. What is the solid content of a frustum of a cone, its height being 30 feet, the greater diameter 6 feet, and less 3 feet? Ans. 494.802.

3. What is the solid content of a frustum of a cone, the diameter of the greater end being 5 feet, and that of the less end 2 feet 6 inches, and its altitude 20 feet?

Ans. 229.075. 4. The length of a mast is 60 feet, its diameter at the greater end 20 inches, and that of the less end 12 inches, what is its solidity? Ans. 85.521.

5. How many cubic feet are there in a piece of timber, of which the ends are squares, each side of one end being 15 inches, and each side of the other 6 inches, the length along the side being 24 feet? Ans. 19.497.

* 1. A frustum is what is left after a piece has been taken from the vertex of a cone or pyramid, the cutting plane passing parallel to the base, as a b defg, and ABDE.

2. When the ends are circles, add together the squares of their diameters, and the product of their diameters, then multiply the sum by .7854, and again by 3 of the height, for the solidity.

3. To find the surface, multiply the sum of the perimeters of the two ends by the slant height, and half the product added to the areas of the ends, gives the whole surface.

6. What is the solidity of a frustum of an octagonal pyramid, its height being 9 feet, the side of the greater end 30 inches, and that of the less end 20 inches ?

Ans. 191.125. 7. Required the surface and solidity of a frustum of a cone, the diameters being 20 and 6, and slant height 25. Ans. surf. 1363.4544, sol. 3493.4592.

PROBLEM 5.-To find the solidity of a prismoid.

RULE. To the sum of the areas of the two ends, add four times the area of a section parallel to, and equally distant from both; this sum multiplied by one-sixth of the height, gives the solidity.*

EXAMPLES.

1. The length and breadth of a fishpond at the top are 100, and 48 yards; the length and breadth at the bottom 80 and 36 yards; and perpendicular depth 4 yards: how many cubic yards of earth had come out?

(100+80)=90, and

100 X 48
80 X 36

=

(48+36) = 42.

4800 area of top.

2880 area of bottom.

= 15120 area of 4 times the middle section.

90 X 42 X 4:

=

2. What is the solid content of a block of stone, of which the ends are rectangles, the length and breadth of one end being 8 and 6 feet, and of the other end 6 and 4 feet; and its length 9 feet? Ans. 318 cubic feet.

• This rule gives the true content of all frustums, and of all solids whose parallel sections are similar figures; and is also a good approximation for all other kins of solids. When the sides of the base are equal, the solid is a frustum.

3. What is the cubic content of a mill-hopper, the sides at the top being 60 and 50 inches, at bottom 12 and 10 inches, and its perpendicular depth 4 feet? Ans. 344 feet.

4. What is the cubic content of a trough; the top measuring 36 inches by 30; bottom 30 inches by 24; and depth 20 inches ? Ans. 10.347 cubic feet. 5. What is the solidity of a log of Memel timber, the ends of which are rectangles; the length and breadth of one end being 18 and 15 inches, and the corresponding sides of the other end 14 and 11 inches; its length being 18 feet? Ans. 26.6458 cubic feet.

6. How many gallons of water will a washing-tub hold; its length and breadth at the top being 4 and 2 feet, and at the bottom 2 feet 10 inches, and I foot 4 inches; its depth being 18 inches? Ans. 53.838 gallons.

PROBLEM 6.-To find the solidity of a sphere, or globe.

RULE.-Multiply the cube of the diameter by .5236 = (one-sixth of 3.1416); or, the cube of the circumference by .016887, or the surface by one-sixth of the diameter, for the solidity.

EXAMPLES.

1. What is the solidity of a sphere, the diameter of which is 30 inches?

.5236 X 30 X 30 X 30

.12 X 12 X× 12

2. What is the solidity of which is 3 feet?

3. What is the solidity of a

8.18125 cubic feet. Ans.

a sphere, the diameter of Ans. 14.1372 cubic feet. globe 4 feet in diameter ? Ans. 33.5104 cubic feet.

4. What is the solidity of a sphere, the diameter of which is 15 inches? Ans. 1767.15 cubic inches. 5. What is the solidity of a globe, whose diameter is 3 feet 6 inches? Ans. 22.44935 cubic feet. 6. What is the solidity of the earth, supposing it a sphere whose diameter is 7912 miles ?

Ans. 259333411782.8608 cubic miles.

7. Required the solidity of a sphere, the surface of which is 31416.* Ans. 523600. 8. The gilding of a globe cost £56. 10s. 113d.-198, at 2s. 6d. per foot; required its solidity. Ans. 904.7808 feet.

9. The solidity of a sphere is 113.0976 feet; required the number of square yards in its surface. Ans. 12.5664 yards. 10. What is the solidity of a segment 2 inches high, cut off from a sphere 52 inches diameter ?+ Ans. 318.348in. 11. What is the solidity of a segment 4 feet high, cut off from a sphere 20 feet in diameter? Ans. 435.635 cubic feet. 12. What is the solidity of a segment 2 feet 6 inches high, cut off from a sphere whose diameter is 7 feet 3 inches? Ans. 54.8141 cubic feet.

13. What is the curve surface and solidity of a segment 3 inches high, cut off from a sphere, the diameter of which is 8 inches? Ans. surf. 87.9648, sol. 109.0397.

14. If with a pair of compasses, extended 9 inches, a circle be described upon a globe, the diameter of which is 15 inches, what is the surface and solidity of the segment cut off by that circle? Ans. surf. 254.4696, sol. 522.1716.

15. Required the solid content of a segment of the earth; the circumference of its base being the parallel of latitude 36° 52', supposing the earth a perfect sphere whose diameter is 8000 miles.‡ Ans. 27880652800 cubic miles.

* The surface of a sphere 4 times the area of one of its great circles = convex surface of its circumscribing cylinder square of diameter by 3.1416 = diameter X by the circumference.

A sphere two-thirds of its circumscribing cylinder.

The curve surface of a segment is found by multiplying the circumference of the sphere by the height of the segment; or the square of the chord of half the arc by

3.1416.

To find the solidity of a segment.-From three times the diameter of the sphere, subtract twice the height of the segment: then multiply the remainder by the square of the height, and the product again by .5236. Or, to three times the square of the radius of the base of the segment, add the square of the height; multiply the sum by the height, and the product by .5236 for the solidity.

The complement of 36° 52' is 53° 8', the natural sine of which is .800034; and 1: 4000::.8000: the radius of the segment's base.