Grade: Grade 5 Subject: Mathematics Unit: Volume SAT: Geometry+Trigonometry ACT: Math

Volume of Rectangular Prisms

Learn how to measure the space inside 3D shapes! Volume tells us how much a container can hold.

What is Volume?

Volume = The Space Inside

Volume measures how much space a 3D object takes up, or how much it can hold. Think of filling a box with unit cubes!

Imagine you have a cardboard box. Volume tells you how many small cubes could fit inside that box. We measure volume in cubic units - like cubic inches (in³), cubic feet (ft³), or cubic centimeters (cm³).

Remember: Area measures flat surfaces (2D) in square units. Volume measures 3D space in cubic units. The little "3" in cm³ reminds us we're measuring in three dimensions!

Understanding Rectangular Prisms

A rectangular prism is a 3D shape with 6 rectangular faces. Think of a cereal box, a brick, or a shipping box - they're all rectangular prisms!

Length (l) - how long
Width (w) - how wide
Height (h) - how tall
V = l × w × h
Volume = Length × Width × Height
Alternative Formula: Since l × w gives you the area of the base, you can also write: V = Base Area × Height or V = B × h

Counting Unit Cubes

The easiest way to understand volume is to count unit cubes. A unit cube is a cube that is 1 unit on each side.

Example: A prism that is 4 × 3 × 2

Length = 4 units, Width = 3 units, Height = 2 units

Method 1: Count the cubes

Bottom layer: 4 × 3 = 12 cubes

Top layer: 4 × 3 = 12 cubes

Total: 12 + 12 = 24 cubic units

Method 2: Use the formula

V = l × w × h = 4 × 3 × 2 = 24 cubic units

Bottom Layer (4 × 3 = 12 cubes):

Understanding Cubic Units

We use different cubic units depending on what we're measuring:

Cubic Centimeter
cm³

Small items

Cubic Inch
in³

Small boxes

Cubic Foot
ft³

Rooms, fridges

Cubic Meter
m³

Large spaces

Important: Always include the unit in your answer! "24" is not a complete answer - "24 cubic inches" or "24 in³" is correct.

Worked Examples

1 Example: Find the volume of a box

A shipping box is 12 inches long, 8 inches wide, and 6 inches tall.

Solution:

V = l × w × h

V = 12 × 8 × 6

V = 96 × 6

V = 576 cubic inches (in³)

2 Example: Aquarium volume

An aquarium is 50 cm long, 30 cm wide, and 40 cm tall. How many cubic centimeters of water can it hold?

Solution:

V = l × w × h

V = 50 × 30 × 40

V = 1,500 × 40

V = 60,000 cm³

3 Example: Finding a missing dimension

A box has a volume of 120 cubic feet. It is 6 ft long and 4 ft wide. How tall is it?

Solution:

V = l × w × h

120 = 6 × 4 × h

120 = 24 × h

h = 120 ÷ 24

h = 5 feet

Real-World Applications

📦
Shipping Boxes

Knowing volume helps determine how much can fit in a box and shipping costs.

🏊
Swimming Pools

Volume tells us how much water is needed to fill a pool.

🧊
Refrigerators

Volume in cubic feet tells you how much food storage space you have.

🏠
Room Volume

HVAC systems need to know room volume to properly heat or cool a space.

Volume Calculator

Enter the dimensions of a rectangular prism to calculate its volume!

Calculate Volume

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Click "Calculate" to find the volume!

Practice Problems

Solve these volume problems. Click the correct answer!

Problem 1

A box is 6 in long, 4 in wide, and 5 in tall. What is the volume?

Problem 2

A cube has sides of 7 cm. What is its volume?

Problem 3

A prism has a base area of 24 ft² and height of 8 ft. What is the volume?

Problem 4

A box has volume 240 cm³. If l = 8 cm and w = 6 cm, what is h?

Problem 5

Which has greater volume: a 10×10×10 cube or a 12×8×10 prism?

Problem 6: Word Problem

A swimming pool is 25 meters long, 10 meters wide, and 2 meters deep. How many cubic meters of water does it hold?

Check Your Understanding

Volume is measured in:

The formula for volume of a rectangular prism is:

If you double all three dimensions of a box, the volume:

What We Learned

📦

Volume = Space Inside

How much a 3D shape can hold

📐

V = l × w × h

Length times Width times Height

🧊

Cubic Units

Always write units as cm³, in³, ft³, etc.

🔢

Count Cubes

Volume = layers × cubes per layer

Key Takeaway: Volume is the 3D version of area. Just as area measures flat space (length × width), volume measures 3D space (length × width × height). Always remember to use cubic units!

Next Steps

  • Practice measuring real boxes around your home
  • Try finding the volume of different shaped containers
  • Work on problems where you need to find a missing dimension
  • Move on to learn about composite volumes (combining shapes)!