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Volume of
Prisms

Learn how to calculate the volume of right rectangular prisms using the formula \( V = l \times w \times h \) and interpret your results with cubic units.

Pre-Algebra

What Is Volume?

Volume measures the amount of three-dimensional space an object occupies. Think of it as how much stuff you could fit inside a container: water in a fish tank, cereal in a box, or air inside a room.

\(1\)
\(1\)
\(1\)
\(1\)
\(1\)
\(1\)
 6 unit cubes = 6 cubic units Each square represents the top face of a unit cube

What Is a Rectangular Prism?

A right rectangular prism (also called a cuboid or box shape) is a 3D solid with 6 rectangular faces. All edges meet at right angles. It has three key measurements:

Length \( l \)
How long the base stretches from left to right.
Width \( w \)
How wide the base extends from front to back.
Height \( h \)
How tall the prism rises from bottom to top.
\( l \)
\( w \)
\( h \)

Why Volume Matters

Volume shows up everywhere in real life. Whenever you need to know how much space something takes up or how much it can hold, you are thinking about volume.

Packing
How many items fit in a shipping box or suitcase?
Filling
How much water fills a swimming pool or fish tank?
Building
How much concrete is needed to pour a foundation?

The Volume Formula

For any right rectangular prism, the volume equals:

5 cm 3 cm 4 cm

Why the Formula Works

The formula \( V = l \times w \times h \) comes from stacking layers of unit cubes. Here is the reasoning:

1
Build one row. Place cubes along the length. If \( l = 5 \), that is 5 cubes in one row.
2
Fill one layer. Copy the row across the width. If \( w = 3 \), you get \( 5 \times 3 = 15 \) cubes in one flat layer. This is the base area.
3
Stack layers up. Repeat the layer for the height. If \( h = 4 \), you stack 4 layers: \( 15 \times 4 = 60 \) cubes total.

Volume vs. Surface Area

Students often confuse these two measurements. They answer different questions about the same 3D shape:

Volume Surface Area
Measures Space inside Total area of outside faces
Units Cubic (cm\(^3\), in\(^3\)) Square (cm\(^2\), in\(^2\))
Formula \( l \times w \times h \) \( 2(lw + lh + wh) \)
Real-life use Filling a pool Wrapping a gift

How to Find Volume

1
Identify the shape. Confirm it is a right rectangular prism (box shape with right angles).
2
Find the three dimensions. Read or measure the length, width, and height. Check that they share the same unit.
3
Multiply. Compute \( V = l \times w \times h \). You may multiply in any order.
4
Write the answer with cubic units. If dimensions are in inches, the volume is in \(\text{in}^3\). Always label your answer.

Example 1: Shoe Box

A shoe box measures 12 inches long, 6 inches wide, and 5 inches tall. What is its volume?

12 in 6 in 5 in

Example 2: Fish Tank

A fish tank is 50 cm long, 25 cm wide, and 30 cm tall. How many cubic centimeters of water can it hold?

50 cm 25 cm 30 cm

Example 3: Fractional Dimensions

A storage container is \( 2\frac{1}{2} \) ft long, \( 1\frac{1}{2} \) ft wide, and \( 3 \) ft tall. Find its volume.

\( 2\frac{1}{2} \text{ ft} \)
\( 1\frac{1}{2} \text{ ft} \)
3 ft

Example 4: Find a Missing Dimension

A box has a volume of 240 cm\(^3\). Its length is 10 cm and width is 6 cm. What is the height?

1
Write the formula: \( V = l \times w \times h \)
2
Substitute known values: \( 240 = 10 \times 6 \times h \)
3
Simplify: \( 240 = 60h \)
4
Solve: \( h = \frac{240}{60} = 4 \text{ cm} \)

The Base Area Shortcut

Sometimes a problem gives you the area of the base directly instead of separate length and width. In that case, use the equivalent formula:

\(B\)
\(h\)
 Shaded face = Base area (B)

Units Matter

The unit you use for the dimensions determines the cubic unit for your answer. Keeping track of units is a critical skill:

Dimension Unit Volume Unit Example
centimeters (cm) cm\(^3\) Small boxes, blocks
inches (in) in\(^3\) Shoe boxes, books
feet (ft) ft\(^3\) Rooms, furniture
meters (m) m\(^3\) Swimming pools, buildings

Common Mistakes

✕ Adding Instead of Multiplying
Writing \( V = l + w + h \) instead of \( V = l \times w \times h \). Volume requires multiplication, not addition.
✕ Wrong Units
Writing cm\(^2\) (square) instead of cm\(^3\) (cubic). Volume is three-dimensional, so the exponent must be 3.
✕ Forgetting to Convert
Multiplying 2 ft \(\times\) 6 in \(\times\) 3 ft without converting inches to feet first. Mixed units produce an incorrect answer.

Question 1 of 5

A rectangular prism has dimensions \( l = 8 \text{ cm} \), \( w = 5 \text{ cm} \), and \( h = 3 \text{ cm} \). What is its volume?

Question 2 of 5

A storage crate has a base area of \( 48 \text{ ft}^2 \) and a height of \( 6 \text{ ft} \). What is the volume of the crate?

Question 3 of 5

A box has a volume of \( 360 \text{ in}^3 \). Its length is \( 12 \text{ in} \) and its height is \( 5 \text{ in} \). What is the width?

Question 4 of 5

A rectangular prism has dimensions \( 4.5 \text{ m} \), \( 2 \text{ m} \), and \( 3 \text{ m} \). Which is the correct volume with units?

Question 5 of 5

Two boxes have the same height of \( 4 \text{ in} \). Box A has a base of \( 10 \times 3 \) inches. Box B has a base of \( 6 \times 5 \) inches. Which statement is true?