📖 Lesson 19 ⏱ ~30 min Year 10 · Unit 1 ⚡ +50 XP

Volume of Composite Solids

Combine simple solids to solve real-world volume problems. Add, subtract and use Pythagoras to find missing measurements.

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Warm-up

Think First

+5 XP each

Q1: A shed has a rectangular base 5 m by 4 m with walls 2.5 m high, and a triangular-prism roof on top. Before calculating, guess the total volume. What strategy will you use?

Q2: A cube of side 10 cm has a cylindrical hole drilled through it. Will the remaining volume be closer to 1000 cm$^3$ or 500 cm$^3$? Make a prediction and explain.

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Intentions

Learning Intentions

Know

That composite solids are made by combining or removing simpler solids.
How to apply Pythagoras' theorem to find missing dimensions.

Understand

Why addition and subtraction strategies work for composite volumes.
When to use each strategy for maximum efficiency.

Can Do

Calculate the volume of composite solids using addition and subtraction.
Use Pythagoras' theorem to find missing heights or lengths.

From the lesson

Success Criteria

I can split a composite solid into simpler parts and calculate each volume.
I can calculate the volume of a solid with a hole or cut-out by subtraction.
I can use Pythagoras' theorem to find missing dimensions needed for volume calculations.

From the lesson

Key Terms

Composite solid — A 3D shape made by joining two or more simple solids together, or by removing part of a solid.

Addition strategy — Splitting a composite solid into parts, finding each volume, then adding them.

Subtraction strategy — Finding the volume of an enclosing solid, then subtracting the volume of the missing part.

Cross-sectional area — The area of the face you see when slicing through a solid perpendicular to its length.

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Misconceptions

Common Mistakes to Avoid

Wrong: Adding the volumes of overlapping parts twice. When two solids join, the overlapping region should only be counted once.

Right: Carefully identify the boundaries between parts. Each part's dimensions should only include its own volume.

Wrong: Forgetting to convert all dimensions to the same unit before calculating.

Right: Convert mixed units first. If some dimensions are in cm and others in m, convert everything to one unit.

Concept

Addition Strategy for Composite Solids

+5 XP

When a solid is made by joining simpler shapes, calculate each part's volume separately and add them together.

Steps:

Identify the simple solids that make up the composite shape.
Find all dimensions for each part. Use Pythagoras if needed.
Calculate the volume of each part.
Add the volumes together.

What to write in your book

Addition strategy: split the solid into simpler parts and add volumes
Identify each simple solid in the composite shape
Find all dimensions; use Pythagoras if needed
Calculate each volume, then add them together

A composite solid is made from a rectangular prism 10 cm by 8 cm by 5 cm with a cylinder of radius 3 cm and height 5 cm on top. What is the total volume?

Correct! Prism = $10 \times 8 \times 5 = 400$ cm$^3$. Cylinder = $\pi(3)^2(5) = 45\pi$ cm$^3$.

Not quite. Prism = $10 \times 8 \times 5 = 400$ cm$^3$. Cylinder = $\pi(3)^2(5) = 45\pi$ cm$^3$.

From the lesson

Worked Example 1

Worked Example 1 — Addition Strategy

Given: A toy consists of a rectangular block 8 cm by 6 cm by 4 cm with a cylinder of radius 2 cm and height 6 cm on top. Find the total volume.

Method: Block: $V = 8 \times 6 \times 4 = 192$ cm$^3$. Cylinder: $V = \pi(2)^2(6) = 24\pi \approx 75.4$ cm$^3$.

Answer: Total volume = $192 + 24\pi \approx \mathbf{267.4}$ cm$^3$ (1 d.p.)

Concept

Subtraction Strategy for Composite Solids

+5 XP

When a solid has a hole, tunnel or cut-out, calculate the volume of the original solid and subtract the volume of the missing part.

Steps:

Find the volume of the complete enclosing solid.
Find the volume of the hole or missing part.
Subtract: $V_{composite} = V_{enclosing} - V_{hole}$.

What to write in your book

Subtraction strategy: find enclosing volume minus hole volume
$V_{composite} = V_{enclosing} - V_{hole}$
Use when the solid has a hole, tunnel or cut-out

A cube of side 10 cm has a cylindrical hole of radius 2 cm drilled through its entire height. What is the remaining volume?

Correct! Cube = $10^3 = 1000$ cm$^3$. Cylinder = $\pi(2)^2(10) = 40\pi$ cm$^3$.

Not quite. Cube = $10^3 = 1000$ cm$^3$. Cylinder = $\pi(2)^2(10) = 40\pi$ cm$^3$.

From the lesson

Worked Example 2

Worked Example 2 — Subtraction Strategy

Given: A concrete block measures 2 m by 1.5 m by 0.8 m with a cylindrical hole of diameter 0.4 m drilled through its full height. Find the volume of concrete.

Method: Block: $V = 2 \times 1.5 \times 0.8 = 2.4$ m$^3$. Hole: $r = 0.2$ m, $V = \pi(0.2)^2(0.8) = 0.032\pi \approx 0.1005$ m$^3$.

Answer: Concrete = $2.4 - 0.032\pi \approx \mathbf{2.30}$ m$^3$ (2 d.p.)

From the lesson

Interactive

Interactive: Composite Volume Builder

From the lesson

Practice

Your Turn

Question 1: A solid is made from a cube of side 6 cm with a smaller cube of side 2 cm removed from one corner. Find the remaining volume.

Question 2: A storage shed is a rectangular prism 10 m by 6 m by 3 m with a half-cylinder roof of radius 3 m along the 10 m length. Find the total volume.

Question 3: Explain when you would use subtraction rather than addition for composite solids.

From the lesson

Revisit

Revisit Your Thinking

Look back at your Think First answer about the shed with the roof. Calculate the exact total volume. The base is a rectangular prism 5 m by 4 m by 2.5 m, and the roof is a triangular prism with base 5 m, height 1.5 m and length 4 m. Was your strategy correct? Did you need to use Pythagoras?

Reflect

Revisit your thinking

reflect

Earlier you were asked: What was your first thought on this topic?

Now that you've worked through the lesson, write a fuller answer. What changed in your thinking?

Recap

Quick Review

Addition

Split solid; add volumes

Subtraction

Enclose solid; subtract hole

Prism

$V = A_{base} \times h$

Cylinder

$V = \pi r^2 h$

Cone

$V = \frac{1}{3}\pi r^2 h$

Pythagoras

$a^2 + b^2 = c^2$

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Real-Life Link

Australian grain silos are iconic composite solids — a cylinder with a conical roof. Farmers calculate total capacity to know how much wheat, barley or canola they can store. The Port of Melbourne handles millions of tonnes of grain annually, and accurate volume calculations ensure ships are loaded efficiently. In construction, concrete footings for buildings often have cylindrical piers extending from rectangular slabs — another composite solid. Excavators calculate cut-and-fill volumes by breaking irregular terrain into prisms and summing their volumes.

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Game

Game Time!

Test your composite volume skills in an interactive challenge.

Play Composite Volume Challenge

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