2. Required the solidity of a 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.

DEFINITIONS.

10. A WEDGE is a solid bounded by five planes: viz., a rectangle, ABCD, called the base of the wedge; two trapezoids ABHG, DCHG, which are called the sides of the wedge, and which intersect each other in the edge GH; and the two triangles

D

GDA, HCB, which are called the ends of the wedge.

H

When AB, the length of the base, is equal to GH, the trapezoids ABHG, DCHG, become parallelograms, and the wedge is then one-half the parallelopipedon described on the base ABCD, and having the same altitude with the wedge. The altitude of the wedge is the perpendicular let fall from any point of the line GH, on the base ABCD.

11. A RECTANGULAR PRISMOID is a solid resembling the frustum of a quadrangular pyramid. The upper and lower bases are rectangles, having their corresponding sides parallel, and the convex surface is made up of four trapezoids. The altitude of the prismoid is the perpendicular distance between its bases.

Suppose AB, the length of

D

M

G

H

B

the base, to be equal to GH, the length of the edge, the solidity will then be equal to half the parallelopipedon,

wedge will then be divided into the triangular prism BCH-G, and the quadrangular pyramid G-AMND. Then, the solidity of the prism

=

bhl; the solidity of the pyramid = } bh (L − 1); and their sum,

{bhl + } bh(L − 1) = } bh3l + } bh 2 L — } bh2l = } bh(2Ľ+i).

If the length of the base is less than the length of the edge, the solidity of the wedge will be equal to the differ ence between the prism and pyramid, and we shall have for the solidity of the wedge,

† bhl — } bh(l — L) = | bh3l — } bh21 +‡bh2L = } bh(2L + 1). Ex. 1. If the base of a wedge is 40 by 20 feet, the edge 35 feet, and the altitude 10 feet, what is the solidity? Ans. 3833.33.

2. The base of a wedge being 18 feet by 9, the edge 20 feet, and the altitude 6 feet, what is the solidity?

Ans. 504.

12. To find the solidity of a rectangular prismoid. Add together the areas of the two bases and four times the

area of a parallel section at equal distances from the bases: then multiply the sum by one-sixth of the altitude.

For, let L and B denote the length and breadth of the lower base, 7 and b the length and breadth of the upper base, M and m the length and breadth of the section equidistant from the bases, and h the altitude of the prismoid.

Through the diagonal edges L

B

M

L

and let a plane be passed, and it will divide the pris moid into two wedges, having for bases, the bases of the prismoid, and for edges the lines L and '

=

1.

The solidity of these wedges, and consequently, of the prismoid, is

} Bh(2L + 1) + † bh(21 + L) = } h(2BL + Bl + 2b1 + bL) }h(BL+ Bl+bL + bl + BL + bl).

=

But since M is equally distant from L and 7, we have, 2ML+1, and 2m B+b;

=

=

hence, 4Mm (L + 1) × (B + b) = BL + Bl + bL + bl. Substituting 4Mm for its value in the preceding equa tion, and we have for the solidity

†h(BL+bl + 4Mm).

REMARK. This rule may be applied to any prismoid whatever. For, whatever be the form of the bases, there may be inscribed in each the same number of rectangles, and the number of these rectangles may be made so great that their sum in each base will differ from that base, by less than any assignable quantity. Now, if on these rectangles, rectangular prismoids be constructed, their sum will differ from the given prismoid by less than any assignable quantity. Hence, the rule is general.

Ex. 1. One of the bases of a rectangular prismoid is 25 feet by 20, the other 15 feet by 10, and the altitude 12 feet; required the solidity. Ans. 3700.

2. What is the solidity of a stick of hewn timber, whose ends are 30 inches by 27, and 24 inches by 18, its length being 24 feet? Ans. 102 ft.

OF THE MEASURES OF THE THREE ROUND BODIES.

13. To find the surface of a cylinder.

Multiply the circumference of the buse by the altitude, and the product will be the convex surface (B. VIII., P. 1). To this add the areas of the two bases, when the entire surface is required.

Ex. 1. What is the convex surface of a cylinder, the diameter of whose base is 20, and whose altitude is 50? Ans. 3141.6.

2. Required the entire surface of a cylinder, whose altitude is 20 feet, and the diameter of its base 2 feet. Ans. 131.9472.

14. To find the convex surface of a cone.

Multiply the circumference of the base by half the slant height (B. VIII., P. 3): to which add the area of the base, when the entire surface is required.

Ex. 1. Required the convex surface of a cone, whose slant height is 50 feet, and the diameter of its base 8 feet? Ans. 667.59.

2. Required the entire surface of a cone, whose slant height is 36, and the diameter of its base 18 feet.

Ans. 1272.348.

15. To find the surface of a frustum of a cone.

Multiply the slant height of the frustum by half the sum of the

circumferences of the two bases, for the convex surface (B. VIII., P. 4): to which add the areas of the two bases, when the entire surface is required.

Ex. 1. To find the convex surface of the cone, the slant height of the frustum being 12 circumferences of the bases 8.4 feet and 6 feet.

frustum of a

feet, and the

Ans. 90.

2. To find the entire surface of the frustum of a cone, the slant height being 16 feet, and the radii of the bases 3 feet and 2 feet. Ans. 292.1688.

16. To find the solidity of a cylinder.

Multiply the area of the base by the altitude (B. VIII., P. 2).

Ex. 1. Required the solidity of a cylinder whose alti tude is 12 feet, and the diameter of its base 15 feet.

2. Required the solidity of a cylinder 20 feet, and the circumference of whose inches.

Ans. 2120.58. whose altitude is base is 5 feet 6 Ans. 48.144.

17. To find the solidity of a cone.

Multiply the area of the base by the altitude, and take one-third of the product (B. VIII., P. 5).

Ex. 1. Required the solidity of a cone whose altitude is 27 feet, and the diameter of the base 10 feet.

Ans. 706.86.

2. Required the solidity of a cone whose altitude is 10+ feet, and the circumference of its base 9 feet.

Ans. 22.56.

18. To find the solidity of a frustum of a cone.

Add together the areas of the two bases and a mean propor tional between them, and then multiply the sum by one-third of the altitude (B. VIII., P. 6).

Ex. 1 To find the solidity of the frustum of a cone, the altitude being 18, the diameter of the lower base 8, and that of the upper base 4. Ans. 527.7888.

2. What is the solidity of the frustum of a cone, the altitude being 25, the circumference of the lower base 20, and that of the upper base 10? Ans. 464.216.

3. 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 40 inches, how many gallons of wine will it contain, there being 231 cubic inches in a gallon?

Ans. 79.0613.

19. To find the surface of a spherical zone.

Multiply the altitude of the zone by the circumference of a great circle of the sphere, and the product will be the surface (B. VIII., P. 10, c. 2).

Exc. 1. The diameter of a sphere being 42 inches, what is the convex surface of a zone whose altitude is 9 inches? Ans. 1187.5248 sq. in.

2. If 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.