NOTE.-The surface of a zone may be found by multiplying its height by the whole circumference of the sphere.

PROBLEM II.

732. To find the solidity or volume of a sphere.

RULE.-Multiply the surface by of the diameter, or of the cube of the diameter by 3.1416.

The solidities of spheres are to each other as the cubes of their radii or diameters.

EXAMPLES.

1. What is the solidity of a sphere whose diameter is 18 inches? 2. The diameter of a globe is 2 feet 6 inches: what is its volume or solidity?

3. If the diameter of the earth be 7912 miles, how many cubic miles does it contain?

4. Required the diameter of a globe which contains 6 times as many cubic inches as a globe 6 inches in diameter.

NOTE.-The volumes of all similar solids are to each other as the cubes of their like dimensions.

MISCELLANEOUS EXAMPLES.

1. A board, in the form of an isosceles triangle, is 14 feet long, and 2 feet wide at one end. What must be the length of a piece, to be cut from the wide end, which will contain 6 square feet?

2. What is the area of the space included between two concentric circles, one of which is 12 inches, and the other 8 inches in diameter ?

3. The diagonal of a rectangle is 20 feet long, and its base 16 feet: what is its area?

4. Find the area of a parallelogram, and also of a triangle, whose base is 9 feet, and altitude 12 feet.

5. Find the solidity of a prism, of which the area of the base is 48 square inches, and the altitude 24 inches.

MEASUREMENT OF LUMBER.

733. The following practical methods of calculating the contents of lumber are generally employed by lumber dealers.

CASE I.

734. To find the superficial contents of a board one inch thick.

EXAMPLE. How many square feet are there in a board 12 feet long, and 9 inches wide?

12 × 9 12

= 9, Ans.

foot, 12 x = 9.

ANALYSIS.-The contents equal the length in feet multiplied by the breadth in inches and divided by 12; or, since 9 inches equal of a

It will be observed that, in any case, if one of the dimensions be given in feet, and the other in inches, their product must always be divided by 12. Therefore, 12 may be cancelled from either factor, before multiplying. Hence, we have the following

RULE. When the length is given in feet, and the breadth in inches, divide either dimension by 12, and multiply the quotient by the other; the product will be the contents in square feet.

GENERAL APPLICATIONS.

1. A board 12 feet long and 12 inches wide, contains 12 square feet. 2. A board 12 feet long and of any width, contains as many square feet as it is inches wide.

3. A board of any length contains as many square feet as it is inches in width, increased or diminished in the same proportion as its length is greater or less than 12 feet.

4. A board of any width contains as many square feet as it is feet in length, increased or diminished in the same proportion as its width is greater or less than 12 inches.

EXAMPLES.

1. How many square feet are there in a board 8 feet long, and 18 inches wide?

8+ 4 = 12; or 186 12; or of 1812. Ans. 12.

=

2. Find the contents of a board 15 feet long, and 8 inches wide.

3. Required the contents of a board 10 feet long, and 9 inches wide.

4. Calculate in the same manner the contents of the following:

NOTE.—Since the number of boards, the length, and the breadth, are all factors of a product to be divided by 12, we may divide any one of them by 12, before multiplying.

Thus, in the first example, cancel 12, and multiply 16 by 9. In the second example, multiply 10 by 10, and add to the product of itself, because 14 is greater than 12. In the third example, increase 20 by of itself and multiply by 9. An expert will perform such calculations, mentally, with great rapidity.

CASE II.

735. To find the contents of planks, scantlings, joists, and square timber.

Lumber of this description is usually reduced to its equivalent in boards one inch thick. A plank 13 inches wide is measured as a board 26 inches wide, and a piece of scantling 3 by 2 inches, as a board 6 inches wide. Hence, we have the following

RULE.-Multiply the width by the thickness, and then proceed as in Case I.

736. For any lumber 3 inches thick:

RULE.-Multiply the width by one fourth of the length; or, half the length by half the width.,

737. For any lumber 2, 3, 4, or 6 inches thick:

RULE.-Multiply the width by the same part of the length that the thickness is of 12.

738. For square timber:

RULE.-Multiply the length by one twelfth of the area of the end in inches.

NOTES.-1. In the above rules, understand the width to be in inches, and the length in feet.

2. When the area of the end of a stick of timber is 12 square inches, the length is its contents in square feet, board measure.

3. If the area of the end be 144 square inches, the contents will be 12 times the length, or one square foot for every inch in length.

EXAMPLES.

1. Find the contents, in board measure, of a plank 16 feet long, 8 inches wide, and 2 inches thick.

165 (of 16) = 211

or, 8 x 2

(of 16) = 213.

Ans. 211 feet.

2. Find the contents of a plank 14 feet long, 10 inches wide, and 3 inches thick.

7 X 5 = 35.

Ans. 35 feet.

3. Required the contents of 10 pieces of scantling, each 14 feet long, 3 inches wide, and 2 inches thick.

10 X 770.

Ans. 70.

4. How many feet are there in 12 pieces of scantling, 31 by 24, and 16 feet long?

5 16+ 12

2 X 2 X 12 X=140.

Ans. 140.

5. Find the contents, in board measure, of a stick of square timber, 12 × 8, and 18 feet long.

96 × 11

=

144; or 18 X 8 = 144.

Ans. 144 feet.

6. Required the contents of a stick of timber, 8 x 4, and 15

NOTE.-The above examples are given in the form of the items of a Bill. The sign of multiplication is placed between the side dimensions, and omitted before the length, as is customary in practice.

739. To find the cubical contents of square timber.

RULE.-Multiply the area of one end in inches by the length in feet, and divide the product by 144.

EXAMPLES.

1. Find the cubical contents of a stick of square timber 24 feet long, and 16 inches square.

SOLUTION.

16 X 16 X 24

144

= 42 feet, Ans.

2. Required the cubical contents of a stick of timber 26 feet long, and 17 by 15, side dimensions.

3. What are the cubical contents of a stick of timber 14 inches square, and 20 feet long?

4. Find the cubical contents of a stick of timber, 16 × 18, and 28 feet long.

ROUND TIMBER.

740. Round Timber is now very generally bought and sold by its cubical measurement, when reduced to square timber.

The following rule for finding the contents is extensively used, and is considered practically just to both buyer and seller.

RULE.-Deduct from the mean diameter in inches one third of itself; multiply the square of the remainder by the length in feet, and divide the product by 144.

NOTE.-The mean diameter is found by taking one half of the sum of the diameters of the two ends.