01 / 21

Rectangular Prism
Volume

Learn to calculate the volume of rectangular prisms using \( V = l \times w \times h \) and master this essential skill for the SSAT.

SSAT Middle Level: Quantitative

What Is a Rectangular Prism?

length (\( l \)) height (\( h \)) width (\( w \)) 6 faces all rectangles

A rectangular prism is a 3D shape with 6 rectangular faces, 12 edges, and 8 vertices. Think of boxes, bricks, and books: they are all rectangular prisms.

What Is Volume?

3 cubes across 2 high 2 deep = 12 unit cubes Counting Unit Cubes

Volume measures how much space a 3D shape occupies. We measure volume in cubic units: imagine filling the shape with tiny unit cubes, then counting them all.

Why Volume Matters

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

The Volume Formula

\( l \) \( h \) \( w \) Volume = \( l \times w \times h \)

Why Does \( l \times w \times h \) Work?

1
Find the base area: \( l \times w \) gives the number of unit cubes in one flat layer (the bottom face).
2
Stack layers: Multiply the base area by the height \( h \) to count all the layers stacked on top of each other.
3
Result: \( V = \text{(base area)} \times h = l \times w \times h \), the total number of unit cubes.

Area vs. Volume

Area Volume
Measures Flat surface covered Space inside a 3D shape
Dimensions 2D (length × width) 3D (length × width × height)
Units \(\text{cm}^2, \text{in}^2\) \(\text{cm}^3, \text{in}^3\)
Example Floor of a room Entire room

Step-by-Step Approach

1
Identify the shape: confirm it is a rectangular prism (all faces are rectangles).
2
Find the three dimensions: read or measure \( l \), \( w \), and \( h \). Make sure all units match.
3
Multiply: compute \( V = l \times w \times h \). Multiply any two first, then multiply by the third.
4
Label your answer with cubic units (e.g., \(\text{cm}^3\)).

Example 1: Basic Volume

5 cm 4 cm 3 cm \( V = l \times w \times h \) \( = 5 \times 3 \times 4 \) \( = 60 \;\text{cm}^3 \)

Example 2: Larger Numbers

12 in 8 in 6 in \( V = l \times w \times h \) \( = 12 \times 6 \times 8 \) \( = 576 \;\text{in}^3 \)

Example 3: Word Problem

A fish tank is 20 inches long, 10 inches wide, and 12 inches tall. What is the volume of water the tank can hold?

1
Identify dimensions: \( l = 20 \), \( w = 10 \), \( h = 12 \) (all in inches).
2
Apply formula: \( V = 20 \times 10 \times 12 \).
3
Calculate: \( 20 \times 10 = 200 \), then \( 200 \times 12 = 2{,}400 \).

Example 4: Decimals

2.5 m 1.2 m 4 m \( V = l \times w \times h \) \( = 2.5 \times 4 \times 1.2 \) \( = 12 \;\text{m}^3 \)

Special Case: The Cube

\( s \) \( s \) \( s \) \( V = s \times s \times s \) \( V = s^3 \)

Order Doesn't Matter

Multiplication is commutative, so you can multiply \( l \), \( w \), and \( h \) in any order and get the same volume. Pick the pair that is easiest to compute first.

Method A
\( 7 \times 5 = 35 \)
\( 35 \times 4 = 140 \)
Method B
\( 5 \times 4 = 20 \)
\( 20 \times 7 = 140 \)
Method C
\( 7 \times 4 = 28 \)
\( 28 \times 5 = 140 \)

Common Mistakes

✕ Forgetting a Dimension
Computing \( l \times w \) instead of \( l \times w \times h \). This gives area, not volume.
✕ Wrong Units
Writing \(\text{cm}^2\) instead of \(\text{cm}^3\). Volume always uses cubic units.
✕ Mixing Units
Using inches for one dimension and feet for another. Convert to the same unit first.

Summary Table

Shape Formula Example Volume
Rectangular Prism \( l \times w \times h \) \( 6 \times 4 \times 3 \) \( 72 \,\text{units}^3 \)
Cube \( s^3 \) \( 5^3 \) \( 125 \,\text{units}^3 \)
Square-base Prism \( s^2 \times h \) \( 3^2 \times 10 \) \( 90 \,\text{units}^3 \)

Question 1 of 5

A box is 8 cm long, 5 cm wide, and 3 cm tall. What is its volume?

Question 2 of 5

What is the volume of a cube with a side length of 6 inches?

Question 3 of 5

A storage container has a volume of \( 360 \,\text{ft}^3 \). Its length is 12 ft and its width is 6 ft. What is its height?

Question 4 of 5

A rectangular prism has dimensions 2.5 m, 4 m, and 6 m. What is its volume?

Question 5 of 5

Two boxes each measure 3 in by 4 in by 5 in. What is the total combined volume of both boxes?