Composite Volume | Open Textbooks

What is a Composite Shape?

Composite = Made of Parts

A composite shape is made up of two or more simpler shapes joined together. Think of an L-shaped building or a staircase - they're made of rectangles combined!

Real objects aren't always simple rectangles. A house might have an attached garage, a swimming pool might have a diving area, or a building might have wings. To find the volume of these shapes, we break them into simpler parts.

Two Strategies:

Addition Method: Break the shape into parts and ADD their volumes
Subtraction Method: Start with a larger shape and SUBTRACT the missing piece

Addition Method: Break It Apart

The most common approach is to divide the composite shape into rectangular prisms, find each volume, then add them together.

1 Identify the separate parts

Look for natural divisions where you can split the shape into rectangles.

2 Find the dimensions of each part

Determine the length, width, and height of each rectangular piece.

3 Calculate each volume

Use V = l × w × h for each part.

4 Add the volumes together

Total Volume = Volume A + Volume B + ...

Example: L-Shaped Room

A

B

Part A: 4 ft × 3 ft × 10 ft = 120 ft³

Part B: 6 ft × 3 ft × 10 ft = 180 ft³

Total Volume = 120 + 180 = 300 ft³

Subtraction Method: Remove the Missing Part

Sometimes it's easier to imagine the full rectangle and subtract what's been removed.

1 Imagine the complete shape

Picture what the shape would look like if it were a complete rectangular prism.

2 Calculate the full volume

Find the volume of the complete rectangular prism.

3 Calculate the missing piece

Find the volume of the part that's cut out or missing.

4 Subtract to find the answer

Total Volume = Complete Volume - Missing Volume

Example: Block with a Hole

Cut Out

Outer Block: 10 cm × 8 cm × 6 cm = 480 cm³

Cut Out: 4 cm × 4 cm × 6 cm = 96 cm³

Total Volume = 480 - 96 = 384 cm³

Which Method Should I Use?

Use Addition When...

The shape clearly splits into 2-3 parts
Parts are different sizes
L-shapes, T-shapes, staircases
Buildings with wings or extensions

Use Subtraction When...

There's a hole or cutout
A corner is missing
The shape is "almost" a rectangle
Only a small piece is removed

Pro Tip: Both methods should give you the same answer! If you're unsure, try both ways and check that your answers match.

Worked Examples

Example 1: T-Shaped Building

A T-shaped building has these dimensions:

Top bar: 20 m long, 8 m wide, 12 m tall
Stem: 6 m long, 15 m wide, 12 m tall

Solution using Addition:

Volume of top = 20 × 8 × 12 = 1,920 m³

Volume of stem = 6 × 15 × 12 = 1,080 m³

Total = 1,920 + 1,080 = 3,000 m³

Example 2: Pool with Wading Area

A swimming pool is 25 m × 10 m × 3 m deep, but has a shallow wading area (5 m × 10 m × 2 m) cut from one end.

Solution using Subtraction:

Full pool volume = 25 × 10 × 3 = 750 m³

Shallow cutout = 5 × 10 × 2 = 100 m³

Actual pool = 750 - 100 = 650 m³

Note: The wading area is 1 m deep, so we subtract the extra 2 m of depth that would have been there.

Example 3: Staircase

A 3-step staircase where each step is 10 in wide, 8 in deep, and 6 in tall.

Solution:

Step 1 (bottom, tallest): 10 × 8 × 18 = 1,440 in³

Step 2 (middle): 10 × 8 × 12 = 960 in³

Step 3 (top): 10 × 8 × 6 = 480 in³

Total = 1,440 + 960 + 480 = 2,880 in³

Real-World Composite Shapes

🏠

Houses

Main house + garage + additions

🏊

Pools

Deep end + shallow end

📦

Packaging

Boxes with inserts or cutouts

🏢

Buildings

Towers, wings, and lobbies

Composite Volume Calculator

Calculate the volume of two rectangular prisms combined!

Add Two Rectangular Prisms

Prism A (Blue)

Length

Width

Height

Prism B (Green)

Length

Width

Height

Enter dimensions and click "Calculate Total Volume"

Practice Problems

Solve these composite volume problems. Click the correct answer!

Problem 1: L-Shape

An L-shaped prism has two parts:
Part A: 5 × 3 × 4 cm
Part B: 3 × 3 × 4 cm
What is the total volume?

Problem 2: Block with Cutout

A block is 10 × 8 × 5 inches.
A 2 × 2 × 5 inch hole is cut through it.
What volume remains?

Problem 3: Two-Story Building

Ground floor: 20 × 15 × 10 ft
Second floor (smaller): 15 × 15 × 10 ft
What is the building's total volume?

Problem 4: T-Shape

Top of T: 12 × 4 × 6 m
Stem of T: 4 × 8 × 6 m
Total volume?

Problem 5: Corner Cut

A 6 × 6 × 6 cube has a 2 × 2 × 2 cube cut from one corner.
What is the remaining volume?

Problem 6: Word Problem

A shipping container is 12 × 8 × 8 feet.
Inside, there's a 4 × 4 × 8 ft refrigeration unit.
How much space is left for other cargo?

Check Your Understanding

When finding volume of a composite shape, you should:

The subtraction method is best when:

If you solve the same problem with addition and subtraction methods, you should get:

What We Learned

🧩

Composite Shapes

Made of multiple simpler shapes combined

➕

Addition Method

Split into parts, add their volumes

➖

Subtraction Method

Full shape minus missing piece

✅

Check Your Work

Both methods should give the same answer

Key Takeaway: Complex shapes aren't scary! Any composite shape can be broken down into simple rectangular prisms. Choose addition when splitting makes sense, subtraction when there's a hole or cutout.

Next Steps

Practice identifying how to split real-world objects
Try solving the same problem both ways to check your work
Look for composite shapes in buildings and furniture around you
Challenge yourself with 3+ part composite shapes!