Volume of Rectangular Prisms

Learn how to calculate the volume of rectangular prisms and cubes using length, width, and height.

Intermediate25 minLesson
Printable worksheet

Definition

The volume of a rectangular prism is the amount of space inside it. We measure volume in cubic units (like cubic centimeters or cubic inches).
A rectangular prism is a 3D shape with 6 rectangular faces. Think of a shoebox, a cereal box, or a brick.

The Volume Formula

V=l×w×hV = l \times w \times h
Where:
  • ll = length (how long)
  • ww = width (how wide)
  • hh = height (how tall)
A cube is a special rectangular prism where all sides are equal:
Vcube=s3V_{\text{cube}} = s^3

Try it now

What is the formula for the volume of a rectangular prism?

Worked Examples

A storage box has length 8 cm, width 5 cm, and height 4 cm. What is its volume?
1

Write the formula

V=l×w×hV = l \times w \times h → Formula identified
2

Substitute the values

V=8×5×4V = 8 \times 5 \times 4 → Values plugged in
3

Multiply length and width

8×5=408 \times 5 = 40 → Base area = 40 cm²
4

Multiply by height

40×4=16040 \times 4 = 160 → V=160V = 160 cm³

Common Mistakes

Confusing volume with surface area

Why it's wrong: Volume measures space inside (cubic units), while surface area measures the outside covering (square units).

Correct: Volume uses multiplication of 3 dimensions: V=l×w×hV = l \times w \times h. Surface area adds up all 6 faces.

Forgetting cubic units

Why it's wrong: Volume is 3-dimensional, so the unit must also be cubed (cm³, m³, in³).

Correct: Always write the unit with a cube: 60 cm³, not 60 cm.

Mixing up measurements from different units

Why it's wrong: Multiplying centimeters by meters gives an incorrect result.

Correct: Convert all dimensions to the same unit first. If length is in meters and width in cm, convert one to match the other.

Interactive Visual

3D Shape Viewer

Faces

6

Edges

12

Vertices

8

Euler's formula: V − E + F = 8 − 12 + 6 = 2

Volume

V = lwh

V = 8 · 5 · 4 = 160

160 cm³

Surface Area

SA = 2(lw + lh + wh)

SA = 2(8 · 5 + 8 · 4 + 5 · 4) = 184

184 cm²

Interactive Sandbox

Expression Calculator

Try these:

History

No calculations yet

Practice Problems

16 problems
Problem 1 of 16
Easy
What is the formula for the volume of a rectangular prism?

Why It Matters

Understanding volume helps you solve everyday problems:
  • Packing: How many toys fit in a box?
  • Shipping: What size box do you need for your package?
  • Aquariums: How much water does a fish tank hold?
  • Construction: How much concrete is needed to fill a foundation?
  • Storage: Will this refrigerator fit in the kitchen space?
Volume calculations are essential in architecture, engineering, manufacturing, and even cooking!

Real World Applications

Shipping and Packaging

Companies calculate box volumes to determine shipping costs and how many items fit in a container.

Example:

A shipping box is 12 in × 8 in × 6 in. Its volume is 12×8×6=57612 \times 8 \times 6 = 576 cubic inches.

1Try It Yourself

You need to ship books in a box measuring 15 cm × 10 cm × 20 cm. Each book is 15 cm × 10 cm × 2 cm.

How many books can fit in the box?

Step 1: Write the mathematical expression

First find the box volume, then the book volume, then divide:

Concrete for Construction

Builders calculate volume to order the right amount of concrete for foundations and slabs.

Example:

A foundation is 10 m × 8 m × 0.5 m. Volume = 10×8×0.5=4010 \times 8 \times 0.5 = 40 m³ of concrete needed.

2Try It Yourself

A sidewalk needs to be 20 m long, 1.5 m wide, and 0.1 m thick.

How many cubic meters of concrete are needed?

Step 1: Write the mathematical expression

Calculate: 20×1.5×0.120 \times 1.5 \times 0.1

Key Takeaways

  • 1Volume measures the space inside a 3D object in cubic units
  • 2Formula for rectangular prism: V=l×w×hV = l \times w \times h (length × width × height)
  • 3Formula for cube: V=s3V = s^3 (side cubed)
  • 4Always include cubic units in your answer (cm³, m³, in³, ft³)
  • 5To find a missing dimension, rearrange the formula and divide

Frequently Asked Questions

They are the same thing! 'Rectangular prism' is common in American English, while 'cuboid' is used in British English. Both refer to a 3D shape with 6 rectangular faces.
They are the same thing! 'Rectangular prism' is common in American English, while 'cuboid' is used in British English. Both refer to a 3D shape with 6 rectangular faces.
Because volume is 3-dimensional (length × width × height). When you multiply cm × cm × cm, you get cm³ (cubic centimeters). Each small cube measures 1 unit on each side.
Yes! Multiplication is commutative, so l×w×h=w×h×l=h×l×wl \times w \times h = w \times h \times l = h \times l \times w. The result is always the same.

Glossary

Volume
The amount of space inside a 3D object, measured in cubic units
Rectangular prism
A 3D shape with 6 rectangular faces (also called a cuboid)
Cube
A rectangular prism where all edges have equal length
Cubic units
Units for measuring volume, like cm³, m³, or in³
Dimensions
The measurements of length, width, and height

Formula Card

Rectangular Prism

V=l×w×hV = l \times w \times h

Volume equals length times width times height

Cube

V=s3V = s^3

Volume equals side cubed (s × s × s)

Today's challengeOne problem, three tries

More in This Topic

Introduction to Volume

Elementary20 min

Metric Volume Units (mL, L)

Elementary20 min

Imperial Volume Units (cups, pints, gallons)

Elementary20 min

Converting Metric Volume Units

Elementary25 min