4.1: Volume

Let’s revisit our friend Wally from “Area and Perimeter” and use another aspect of his yard to introduce the concept of volume. Wally is a swimmer and wants to install a lap pool in his backyard. Because he has some extra space, he is going to build a pool that is 25 yards long, 2 yards wide, and 2 yards deep. How many cubic yards of water must be used to fill the pool (assuming right to the top).

Much as we did with area (counting unit squares), with volume we will be counting unit cubes. What is the volume of a unit cube? Let’s look at figure 1:

Finding the Volume of a Cube

Figure 1. Volume = 1 yd × 1yd × yd = 1 cubic yard

The shape at left is a cube (all sides are equal length). In particular, because all sides are of length 1, this cube is called a unit cube.

We know the area of the base from our previous work (1 yd × 1 yd or 1 square yard). We are going to take that area and extend it vertically through a height of 1 yard so our volume becomes

Volume = 1 yd × 1yd × yd = 1 cubic yard

How does this help Wally? Well, if he can count the number of unit cubes in his pool, he can determine the volume of water needed to fill the pool.

Guided Example

Filling Wally’s Pool

Figure 2. Volume = Length × Width × Height or V = LWH

If we fill the pool with unit cubes, we can fill 25 unit cubes along the length, 2 along the width and 2 along the height. That would give us 25 × 2 × 2 = 100 unit cubes or:

Volume = 25 yd × 2 yd × 2 yd = 100 cubic yd

Explicit formulas for the types of rectangular solids used in the previous section are as follows:

Shape		Volume
Cube of side length L This can also be called a cuboid or rectangular prism.		\(V=L \times L \times L \) \( V = L^3 \)
Box with sides of length L, W, H		\( V = L \times W \times H \)

Notes on Volume

• Volume is a three-dimensional measurement that represents the amount of space inside a closed three-dimensional shape.
• To find volume, count the number of unit cubes inside a given shape.
• If there are units, include units in your final result. Units will always be three-dimensional (i.e. cubic feet, cubic yards, cubic miles, etc…)

Example 1

Find the volume of each shape below.

1. 
2. A box (or cuboid) with sides of length 2 ft, 3 ft, 2.5 ft.

Solutions

1. 64 in³
2. 15 ft³

Volume of A Circular Cylinder

Figure 3.

Can we use what we know about the area of a circle to formulate the volume of a can (also called a cylinder)? Take a look at figure 3.

The base circle is shaded. If we take the area of that circle ( \(A  = \pi r^2 \) ) and extend it up through the height h, then our volume for the can would be:

\(V= \pi r^2 h\).

Example 2

Find the volume of the cylinder shown below.

Solution

432π cm³

Volume of Other Shapes

The chart below shows the volumes of some other basic geometric shapes.

Shape	Volume
Sphere	\(V = \frac{4}{3} \pi r^3 \)
Cone	\( V = \frac{1}{3} \pi r^2 h \)
Pyramid	\( V = \frac{1}{3}LWH \)

Example 3

Find the volume of a sphere with radius 5 meters.

Solution

\( \dfrac{500}{3} \text{cm}^3 \)

Example 4

Determine the volume of each of the following. Include a drawing of the shape with the included information. Show all work. As in the examples, if units are included then units should be present in your final result. Use 3.14 for π and round answers to tenths as needed.

1. Find the volume of a cube with side 3.25 meters.
2. Find the volume of a box with sides of length 4 feet by 2.5 feet by 6 feet.
3. Find the volume of a can with radius 4.62 cm and height 10 cm.
4. Find the volume of a sphere with diameter 12 yards.

Solution

1. 34.3 m³ or 34.3 cubic meters
2. 60 ft³ or 60 cubic feet
3. 670.2 cm³ or 670.2 cubic centimeters
4. 904.3 yd³ or 904.3 cubic yards

Example 5

Applications of Volume

If you drank sodas from 5 cans each of diameter 4 inches and height 5 inches, how many cubic inches of soda did you drink? Use 3.14 for \( \pi \) and round to tenths.

Solution

314.0 in³ (rounded)

Surface Area

Surface area provides the amount of area that is needed to cover the outside of a shape.

Shape	Description	Formula
	Cylinder Surface Area This shape is an extension of a circle. The top and bottom of this shape are circles The area of the sides is a rectangular shape with height h and width equal to the circumference of the circle top/bottom.	\(SA = 2\pi r^2 + 2 \pi rh \)
	Cuboid/Rectangular Prism Surface Area This shape is an extension of a rectangle The sides parallel to each other have the same size.  The prism has 6 sides total and 3 dimensions to measure.	\(SA = 2 (l \cdot w) + 2 (l \cdot h) + 2 (w\cdot h) \)
	Pyramid (4 sided) Surface Area This shape is an extension of a square and triangles. The  bottom of this shape is a square. The 4 sides of this shape have matching triangles that all fit together to form the pyramid	\(SA = b^2+ 2bh \) where \(h\) is the diagonal height of the triangle, not the height to the tip of the pyramid.
	Sphere Surface Area This shape is an extension of a circle. The sphere's surface is measured with a formula The formula is derived from Calculus, but that is outside the scope of this text.	\(SA = {4} \pi r^2 \) where\(r\) is the radius of the sphere.

 

CC licensed content, Shared previously

• Basic Arithmetic Student Workbook. Authored by: Donna Gaudet, Amy Volpe, and Jenifer Bohart. Provided by: Scottsdale Community College. Located at: https://sccmath.wordpress.com/. License: CC BY-SA: Attribution-ShareAlike