GED Mathematical Reasoning: Volume

The measure of VOLUME is stated using CUBED units, since a rectangular solid has three dimensions – length, width, and height.

One cubic foot, for example, is a cube that has a length of one foot, a width of one foot and a height of one foot. The shorthand way of writing “cubic feet” is using the abbreviation for feet, which is lowercase “ft,” along with an exponent of 3.

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1 ft^3

 

Cube

To calculate the volume of a cube, the process is the same – we multiply the length times the width times the height. But since the length and width and height are the same, we can label the sides using only one variable.

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If we let s equal the side measure of a cube, then:

Volume of a Cube = Length \times Width \times Height

V = s \times s \times s

V= s^3

 

Rectangular prism

To find the volume of a rectangular prism, multiply the length, width, and height.

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This calculation can be represented by the formula:

Volume (V) of a rectangular prism = Length \times Width \times Height

V = LWH

 

Cylinder

To calculate the volume of a cylinder, we’ll need to know its height, represented by the letter “h”, and the radius of the circular base, denoted by the letter “r.” These two measures are labeled on the graphic showing on the screen.

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The formula for the volume of a cylinder is:

= \pi \times radius^2 \times height

V = \pi r^2 h

 

Sphere

To calculate the volume of a sphere we’ll need to know the radius, denoted by the letter “r.”

meas81The formula for the volume of a sphere is:

 = \frac{4}{3} \times pi \times radius^3

V = \frac{4}{3}\pi r^3

 

Pyramid

The volume of a pyramid is equal to one-third the volume of the rectangular solid having the same base and height. So it makes sense that the formula for the volume of a pyramid is:

= \frac{1}{3} \times length \times width \times height

V = \frac{1}{3}LWH

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Cone

To calculate the volume of a cone, we’ll need to know its height, represented by the letter h, and the radius of the circular base, denoted by the letter r.

meas88The volume of a cone is equal to one-third the volume of a cylinder with the same base and height, so it makes sense that the formula for the volume of a cone is:

=\frac{1}{3} \times \pi \times radius^2 \times height

V = \frac{1}{3}\pi r^2 h

Before we work through a couple of examples, I would like to make a suggestion.

When finding the volume of a sphere, pyramid, or cone calculate the product of the non-fraction numbers first. Then, as a last step, multiply by the fraction.

We can do this because when multiplying numbers, the order doesn’t matter. And it will make the process a bit easier.

 

Example 1

A swimming pool has a length of 15 feet, a width of 10 feet, and a depth of 5 feet. How much water will the pool hold?

 

When we fill a swimming pool with water, we fill the space enclosed by the walls of the pool. So to calculate the amount of water the swimming pool will hold, we will find the volume.

To find the volume of a rectangular prism, multiply the length, width, and height.

Volume (V) of a rectangular prism = Length \times Width \times Height

V = LWH

So to find the volume of the swimming pool, and consequently the amount of water it will hold, we’ll multiply the length of 15 feet by the width of 10 feet by the height of 5 feet.

V = 15 \times 10 \times 5
V = 750

So we will write the volume as: 750ft^3

 

The fact that the swimming pool measures 750 cubic feet means that it takes up a space the size of 750 of those cubed units. In other words, we could fill the pool with 750 blocks having a length of one foot, a width of one foot and a height of one foot.

 

Example 2

Find the volume of a sphere with a radius of 2 inches.

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The formula for the volume of a sphere is:

V = \frac{4}{3}\pi r^3

This volume formula requires the radius measure, which is 2 inches, so we’ll substitute 2 for r , 3.14 for \pi, and then simplify.

V = \frac{4}{3}(3.14) {(2)}^3
V = \frac{4}{3}(3.14)(8)
V = \frac{4}{3}(25.12)
V = 33.49

Therefore, the volume of the sphere is about 33.49in^3

 

Example 3

Find the volume of the cone shown.

math 201

 

Let’s first write the formula for the volume of a cone.

V = \frac{1}{3}\pi r^2 h

This volume formula requires the radius measure, which is shown to be 3 centimeters. It also requires the height of the cone, which is labeled to be 8 centimeters.

So we’ll substitute 3 for “r”, 8 for “h”, 3.14 for pi and then simplify.

V = \frac{1}{3}(3.14){(3)}^2 (8)
V = \frac{1}{3}(3.14)(9)(8)
V = \frac{1}{3}(3.14)(72)
V = \frac{1}{3}(226.08)
V = 75.36

226.08 times one third (which is the same as dividing by three) is equal to approximately: 75.36

Therefore, the volume of the cone is about 75.36cm^3

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