The Measure of Solids, or Cubic Measure
Just as the rectangle is the chief surface considered in arithmetic, so the rectangular solid is the chief solid body.
A rectangular solid is bounded by six rectangular surfaces, each opposite pair of rectangles being equal and parallel to each other.
A rectangular solid thus has three dimensions—length, breadth, and thickness.
If the length, breadth, and thickness are all equal to one another, the solid is called a cube. Hence, a cubic foot, the unit of volume, is a solid body whose length, breadth, and thickness are each a linear foot. Similarly, a cubic inch measures one linear inch in length, breadth, and thickness; and a cubic yard measures one linear yard in length, breadth, and thickness.
The number of cubic feet (or inches, or yards) in the volume of a rectangular solid is equal to the number of linear feet (or inches, or yards) in the length, multiplied by the number of linear feet (or inches, or yards) in the breadth, multiplied by the number of linear feet (or inches, or yards) in the thickness.
This is usually abbreviated into
Length × breadth × thickness = volume, or cubic content.
For example, suppose the solid in the diagram is 10 feet in length, 8 feet in breadth, and 5 feet in thickness. It is clear that the solid can be cut into five slices, each 1 foot thick, by planes parallel to the bottom. But, the bottom contains 10×8 square feet and above each square foot there is a cubic foot. Thus, each slice contains 10×8 cubic feet. Therefore, since there are five slices, the whole solid contains 10×8×5, or 400 cubic feet.
Since length × breadth × thickness = cubic content, it follows that, if we know any three of these four quantities, we can find the fourth.
The student should remember that
(a) A cubic foot of water weighs 1000 ounces (avoirdupois) approximately.
(b) A gallon of pure water weighs 10 pounds (avoirdupois).
We have thus a relation between weight, capacity, and cubic content.
Table of Cubic Measure
| 1728 | cubic inches (cu. in.) | = | 1 cubic foot (cu. ft.) |
| 27 | cubic feet | = | 1 cubic yard (cu. yd.) |
| 128 | cubic feet | = | 1 cord (C.) |
| Cubic Yard | Cubic Feet | Cubic Inches | ||
| 1 | = | 27 | = | 46656 |
A cord of wood or stone is a pile 8 feet long, 4 feet wide, and 4 feet high.
A pile of wood 4 feet high, 4 feet wide and 1 foot long makes a cord foot. 8 cord feet = 1 cord.
A perch of stone or masonry is 161⁄2 feet long, 11⁄2 feet thick, and 1 foot high, and contains 243⁄4 cubic feet.
A cubic yard of earth is considered a load.
Brick work is commonly estimated by the thousand bricks.
Bricklayers, masons, and joiners commonly make a deduction of one half the space occupied by windows and doors in the walls of buildings.
In computing the contents of walls, masons and bricklayers multiply the entire distance around on the outside of the wall by the height and thickness. The corners are thus measured twice.
A cubic foot of distilled water at the maximum density, at the level of the sea, and the barometer at 30 inches, weighs 621⁄2 pounds or 1000 ounces avoirdupois.
By actual measurements, it has been found that a bushel, dry measure, contains about 11⁄4 cubic feet. This makes it easy to estimate about how many bushels any bin will hold.
Practical Applications of Cubic Measure
Example: An open tank made of iron 1⁄4 inch thick, is 4 feet long, 2 feet 6 inches broad, and 2 feet deep, outside measurement. Assuming that iron weighs 7.8 times as much as water, find the weight of the tank.
The external volume of the tank = 2 × 21⁄2 × 4 cubic feet = 20 cubic feet.
Since the iron is 1⁄4 inch thick, the inside length is 1⁄2 inch less than the outside, the inside breadth is 1⁄2 inch less than the outside, and the inside depth is 1⁄4 inch less than the outside.
Therefore the interior volume
= 291⁄2 × 471⁄2 × 233⁄4 cubic inches
= 59 × 95 × 9516 cubic inches
= 3327911⁄16 cubic inches
Therefore, volume of iron in the tank
= 20 cubic feet - 3327911⁄16 cubic inches
= 12805⁄16 cubic inches.
But 1 cubic foot of iron weighs as much as 7.8 cubic feet of water, i. e., 7.8 × 1000 ounces, or 7800 ounces.
∴ Weight of tank = 12805⁄16 × 78001728 × 16 pounds.
∴ Weight of tank = 361.199 pounds, Ans.
Example: A wood pile is 8 feet high and 40 feet long. The sticks are 4 feet long. How many cords in it?
Solution: Being 8 feet high, it is 2 cords high. 40 feet in length equal 5 cords in length. Hence, the pile contains 2 × 5 cords, or 10 cords.
To estimate a bin:
(1) Find the number of cubic feet in the bin.
(2) Divide the number of cubic feet by 11⁄4.
(3) The result is the number of bushels.
Example: How many bushels will a bin hold, if its inside measurements are, length 20 feet, width 12 feet, depth 8 feet?
Solution: The number of cubic feet in a bin is 8 × 12 × 20, or 1920.
If 1 bushel contains 11⁄4 cubic feet, in 1920 cubic feet there are as many bushels as 11⁄4 is contained times in 1920, or 1536.
Work:
8 × 12 × 20 = 1920
1920 ÷ 11⁄4 = 1536.
The work may be indicated in this way as well:—
8 × 12 × 20 × 4⁄5 = 1536.
To get the number of heaped bushels of corn in the ear in a crib:
(1) Multiply the length of the crib in inches by the width in inches.
(2) Multiply the product obtained, by the height of the corn in the crib in inches.
(3) Divide the result by 2748.
Example: How much corn in the ear can I put into a crib 12 feet wide, 20 feet long, and 10 feet deep?
Solution: The number of cubic inches in the crib is 144 × 240 × 120, or 4,147,200.
Since 2748 cubic inches hold 1 bushel, 4,147,200 cubic inches hold as many bushels as 2748 is contained times in 4,147,200, or 1509+ bushels.
Work:
144 × 240 × 1202748 = 1509+.