General (1-13)

Computing Volume of Three-Dimensional Solids
Three-dimensional solids have length, width, and height. There are many three-dimensional solids, but the most common are rectangular solids, cubes, cylinders, spheres, and cones. Volume is the amount of space within a solid. Volume is expressed in cubic units. Cubic inches or cubic centimeters are used for small spaces and cubic feet or cubic meters for larger spaces.

 

 

Rectangular Solid
A rectangular solid is a three-dimensional solid with six rectangle-shaped sides. [Figure 1-24] The volume is the number of cubic units within the rectangular solid. The formula for the volume of a rectangular solid is:

Volume = Length × Width × Height = L × W × H

In Figure 1-24, the rectangular solid is 3 feet by 2 feet by 2 feet.
The volume of the solid in Figure 1-24 is = 3 ft × 2 ft × 2 ft = 12 cubic feet.

 

 

 

Example: A rectangular baggage compartment measures 5 feet 6 inches in length, 3 feet 4 inches in width, and 2 feet 3 inches in height. How many cubic feet of baggage will it hold? First, substitute the known values into the formula.

V = L × W × H
   = 5'6" × 3'4" × 2'3"
   = 5.5 ft × 3.33 ft × 2.25 ft
   = 41.25 cubic feet

 

Cube
A cube is a solid with six square sides. [Figure 1-25] A cube is just a special type of rectangular solid. It has the same formula for volume as does the rectangular solid which is Volume = Length × Width × Height = L × W × H. Because all of the sides of a cube are equal, the volume formula for a cube can also be written as:

Volume = Side × Side × Side = S³

 

Example: A large, cube-shaped carton contains a shipment of smaller boxes inside of it. Each of the smaller boxes is 1 ft × 1 ft × 1 ft. The measurement of the large carton is 3 ft × 3 ft × 3 ft. How many of the smaller boxes are in the large carton? First, substitute the known values into the formula.

V = L × W × H
   = 3 ft × 3 ft × 3 ft
   = 27 cubic feet of volume in the large carton

Since each of the smaller boxes has a volume of 1 cubic foot, the large carton will hold 27 boxes.

 

 

 

 

Cylinder
A solid having the shape of a can, or a length of pipe, or a barrel is called a cylinder. [Figure 1-26] The ends of a cylinder are identical circles. The formula for the volume of a cylinder is:

Volume = π × radius² × height of the cylinder = π r² × H

One of the most important applications of the volume of a cylinder is finding the piston displacement of a cylinder in a reciprocating engine. Piston displacement is the total volume (in cubic inches, cubic centimeters, or liters) swept by all of the pistons of a reciprocating engine as they move in one revolution of the crankshaft. The formula for piston displacement is given as:

Piston Displacement = π × (bore divided by 2)² × stroke × (# cylinders) 

 

 

 

The bore of an engine is the inside diameter of the cylinder. The stroke of the engine is the length the piston travels inside the cylinder. [Figure 1-27]

Example: Find the piston displacement of one cylinder in a multi-cylinder aircraft engine. The engine has a cylinder bore of 5.5 inches and a stroke of 5.4 inches. First, substitute the known values in the formula.

 

V = π × r² × h = (3.1416) × (5.5 ÷ 2)² × (5.4)
V = 23.758 × 5.4 = 128.29 cubic inches

The piston displacement of one cylinder is 128.29 cubic inches. For an eight cylinder engine, then the total engine displacement would be:

Total Displacement for 8 cylinders = 8 × 128.29 = 1026.32 cubic inches of displacement 

 

 

 

 

Sphere
A solid having the shape of a ball is called a sphere. [Figure 1-28] A sphere has a constant diameter. The radius (r) of a sphere is one-half of the diameter (D). The formula for the volume of a sphere is given as:

V = 4⁄3 × π × radius³ = 4⁄3 × π × r³ or V = 1⁄6 × πD³

Example: A pressure tank inside the fuselage of a cargo aircraft is in the shape of a sphere with a diameter of 34 inches. What is the volume of the pressure tank?

V = 4⁄3 × π × radius³ = 4⁄3 × (3.1416) × (34⁄2)³
   = 1.33 × 3.1416 × 17³ = 1.33 × 3.1416 × 4913
V = 20,528.125 cubic inches 

 

 

 

 

Cone
A solid with a circle as a base and with sides that gradually taper to a point is called a cone. [Figure 1-29] The formula for the volume of a cone is given as:

V = 1⁄3 × π × radius² × height = 1⁄3 × π × r² × H

 

 

 

 

Units of Volume
Since all volumes are not measured in the same units, it is necessary to know all the common units of volume and how they are related to each other. For example, the mechanic may know the volume of a tank in cubic feet or cubic inches, but when the tank is full of gasoline, he or she will be interested in how many gallons it contains. Refer to Figure 1-37, Applied Mathematics
Formula Sheet, at the end of the chapter for a comparison of different units of volume.