Consider a circle having a diameter of 8 in.

Reversing the process to find the diameter, we have

50.265.7854 = 64 sq. in.

Example:

Diameter

=

√64 = 8 in., Answer.

To get the required velocity of flow in a turbine penstock requires a cross-sectional area of 19.5 sq. ft. What diameter of penstock is needed?

78. Measure of Volume. To measure length we use a unit of length, such as the inch, foot, yard, etc., and to measure area we use a unit of surface

measure, such as the square inch, the square foot, etc. There is still another measure of size known as the measure of volume, or cubi

FIG. 35.-One inch cube.

FT.

-I FT

FIG. 36.-One foot cube.

cal contents. The measure of volume is expressed in the cubic unit, such as the cubic inch, cubic foot, cubic yard, etc.

The measure of length takes account of but one dimension, length; that of area considers two dimensions, length and width; but the measure of volume embraces both length and width and a third dimension, thickness.

Figure 35 shows a small block or cube each side of which measures 1 in. The volume of this block is equal to the unit

of volume called the "cubic inch." Similarly a cube, such as Fig. 36, having each side 1 ft. long, equals the unit of volume called the "cubic foot."

Suppose that we cut a slice 1 in. thick from the bottom of the block in Fig. 36, and divide this slice into 1-in. cubes, as illustrated in Fig. 37. Each small cube then contains 1 cu. in. and there will clearly be 144 cubes; therefore, the volume of the slice

is 144 cu. in. As the slice cut off is but 1 in. thick, it is apparent that the block of Fig. 36 could be cut into 12 such slices, each 1 in. thick and each containing 144 cu. in. From this we see that the total volume of the block is 12 X 144 1728 cu. in. Since the volume of the block is also 1 cu. ft., it 1 cu. ft. must contain 1728 cu. in.

=

is quite evident that

79. Finding the Volume.-A rectangular block is shown in Fig. 38, 10 in. long, 4 in. wide, and 3 in. thick. At one end of

10"

FIG. 38.-Rectangular block with a portion divided into cubes.

the block is indicated an imaginary slice 1 in. thick and cut up into 1-in. cubes. By actual count there are 12 cu. in. in this slice. We at once note the fact, also, that the area of the end surface of the block is 3 X 4 12 sq. in., or the same as the number of cubic inches. Since the block is 10 in. long, it could be cut into 10 such slices and its volume is, therefore, 10 X 12 = 120 cu. in.

=

It is quite plain to be seen that if we multiply the length by the area of the end, we will obtain the same result; thus,

The same thing holds true for any block or bar having a constant cross-section; that is, any block or bar that could be cut up into equal slices as illustrated in Fig. 39. It can easily be seen that

LENGTH

LENGTH

LENGTH

FIG. 39.-Comparative volumes of solids of equal lengths.

the area at the end of each bar in Fig. 39 equals the number of cubic units in a slice of unit thickness. The total volume, then, is the product of this amount times the length of the bar, or

=

Volume Area of end X Length

For rectangular bars, such as the top bar of Fig. 39, the area of the end equals the product of the breadth times the thickness,

and the volume thus equals the product of length times breadth times thickness. If L, B, and T represent length, breadth, and thickness, then for rectangular blocks

Volume = LX BX T

In finding the volume it is necessary that the proper units be used. When multiplying length times breadth times thickness,

Area 20 sq. ft.)

FIG. 40.-Steel plate.

all of these dimensions must be in the same unit, that is, all in inches, or all in feet, etc., and the volume will be cubic inches, cubic feet, etc., respectively. In the same manner, if the volume

15'

૦૩

FIG. 41. Water tank.

is found by multiplying the area by the length, the unit of area must be square inches when the length is inches, or square feet when the length is feet, etc.

Examples:

1. Find the volume of the steel plate in Fig. 40 if the area of the

top surface is 20 sq. ft. and the plate is 3 in. thick.

In the case of a rectangular block, the area of any side may be considered the area of the end, and the length will be the dimension at right angles to that side.

In the case of bodies which have a cylindrical shape (such as a piece of shafting) where the end is circular in shape, the area of the end is found by the rule for the area of a circle.

2. The water tank of Fig. 41 is 15 ft. in diameter and 20 ft. high. How many cubic feet of water will it hold when full? (We find the volume here exactly as for a solid body.)

80. Weights of Metals. The chief uses in the shop for calculations of volume are in finding the amount of material needed to make some object, in finding the weight of some object that cannot be conveniently weighed, or in finding the capacity of some bin or other receptacle. Having obtained the volume of an object, it is only necessary to multiply the volume by the known weight of a unit volume of the material to get the weight of the object. In the case of the plate of which we just obtained the volume, 1 cu. ft. will weigh about 489 lb., so the total weight of the plate will be

The weight of the water in the tank will be the number of cubic feet times 62.4, since 1 cu. ft. of water weighs 62.4 lb.

The following table gives the weights per cubic inch and per cubic foot for the most common metals and also for water: