Mensuration of Solids.

3. If the diameters are 20 and 15, what will be the area included between the circumferences? Ans. 137.445.

4. If the diameters are 16 and 10, what will be the area included between the circumferences? Ans. 122.5224.

5 Two diameters are 21.75 and 9.5; required the area of the circular ring. Ans. 300.6609.

6. If the two diameters are 4 and 6, what is the area of the ring? Ans. 15.708

The mensuration of solids is divided into two parts.

1st, The mensuration of the surfaces of solids: and

2d, The mensuration of their solidities.

We have already seen that the unit of measure for plane surfaces, is a square whose side is the unit of length (Bk. IV Def. 7).

2. A curve line which is expressed by numbers is also referred to an unit of length, and its numerical value is the number of times which the line contains the unit.

If then, we suppose the linear unit to be reduced to a straight line, and a square constructed on this line, this will be the unit of measure for curved surfaces.

square

3. The unit of solidity is a cube, whose edge is the unit in which the linear dimensions of the solid are expressed; and

Mensuration of Solids.

the face of this cube is the superficial unit in which the surface of the solid is estimated (Bk. VI. Th. xiii. Sch).

4. The following is a table of solid measure.

Multiply the perimeter of the base by the altitude and the product will be the convex surface: and to this add the area of the bases, when the entire surface is required (Bk. VI. Th. i).

EXAMPLES

1. Find the entire surface of the regular prism whose base is the regular polygon ABCDE and altitude AF, when each side of the base is 20 feet and the altitude AF, 50 feet.

B

AB+BC+CD+DE+EA=100; and AF-50: then

(AB+BC+CD+DE+EA) × AF convex surface

=

Mensuration of Solids.

which becomes, 100 × 50-5000 square feet; which is the convex surface, For the area of the end, we have

AB2x tabular number-area ABCDE,

that is, 203× tabular number, or 400 × 1.720477–688.1908 –

=

Entire surface 6376.3816

2. What is the surface of a cube, the length of each side being 20 feet? Ans. 2400 sq. ft.

3. Find the entire surface of a triangular prism, whose base is an equilateral triangle, having each of its sides equal to 18 inches, and altitude 20 feet. Ans. 91.949 sq. ft.

4. What is the convex surface of a regular octagonal prism, the side of whose base is 15 and altitude 12 feet?

Ans. 1440 sq. ft.

5. What must be paid for lining a rectangular cistern with lead at 2d a pound, the thickness of the lead being such as to require 71b. for each square foot of surface; the inner dimensions of the cistern being as follows: viz. the length 3 feet 2 inches, the breadth 2 feet 8 inches, and the depth 2 feet 6 inches? Ans. £2 3s 105.

PROBLEM II

To find the solidity of a prism.

RULE.

Multiply the area of the base by the perpendicular height, and the product will be the solidity.

Mensuration of Sol'ds

EXAMPLES.

1. What is the solidity of a reguiar pentagonal prism whose altitude Is 20, and each side of the base 15 feet?

To find the area of the base we have by Problem VIII. page 178.

152-225: and 225×1.7204774=387.107415=

the area of the base: hence,

387.107415×20=7742.1483=solidity.

2. What is the solid contents of a cube whose side is 24 inches? Ans. 13824 solid in.

3. How many cubic feet in a block of marble, of which the length is 3 feet 2 inches, breadth 2 feet 8 inches, and height or thickness 2 feet 6 inches? Ans. 21 solid ft.

4. How many gallons of water, ale measure, will a cistern contain whose dimensions are the same as in the last example ? Ans. 129.

5. Required the solidity of a triangular prism whose altitude is 10 feet, and the three sides of its triangular base 3, 4, and 5 feet. Ans. 60 solid ft.

6. What is the solidity of a square prism whose height is 5 feet, and each side of the base 1 fcot?

Ans 97 solid ft.

Mensuration of Solids.

7. What is the solidity of a prism whose base is an equi. Lateral triangle, each side of which is 4 feet, the height of the prisin being 10 feet? Ans. 69.282 solid ft.

8 What is the number of cubic or solid feet in a regular rentagonal prism of which the altitude is 15 feet and cach side of the base 3.75 feet? Ans. 362.913

PROBLEM III.

To find the surface of a regular pyramid.

RULE.

Multiply the perimeter of the base by half the slant height, and the product will be the convex surface: to this add the area of the base, if the entire surface is required (Bk. VI. Th_vi)