Volume of Composite Shapes
Every question with a fully worked solution.
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Theory
A composite solid is made by joining prisms together, or by cutting a piece out of a prism. To find its volume, split it into prisms and add, take away the missing piece, or use \(V = A \times h\) when the same face goes all the way through. This Year 7 maths page shows all three ways.
Adding. Split the solid into rectangular prisms \(A\), \(B\), ... Find each volume with \(V = l \times w \times h\), then add: \(V = V_A + V_B\). Every way of splitting gives the same volume.
Missing lengths first. A part’s length is often not shown. Find it from the lengths you know: total minus the known part, such as \(6 - 2 = 4\) cm.
Subtracting. For a hole or a cut-out, find the volume of the whole block, then subtract the missing piece: \(V = V_{\text{block}} - V_{\text{hole}}\).
The same face all the way through. If the front face goes straight through to the back, it is the cross-section. Find its area \(A\) (split it into a rectangle and a triangle if needed), then \(V = A \times h\), where \(h\) is the length of the prism.
Built from cubes. Count one layer (or one stack) at a time, and remember the cubes hidden at the back.
Three ways:
How to find the volume of a composite solid
- Find the missing lengths.
- Split into rectangular prisms, or see the whole block and the missing piece, or find the cross-section.
- Find each volume.
- Add or subtract, then check another way.
| \(9 \times 5 \times 2\) | \(=\) | \(90\) |
| \(4 \times 5 \times 3\) | \(=\) | \(60\) |
| \(90 + 60\) | \(=\) | \(150\) |
\(150~\text{cm}^3\)
| \(10 \times 6 \times 5\) | \(=\) | \(300\) |
| \(4 \times 2 \times 6\) | \(=\) | \(48\) |
| \(300 - 48\) | \(=\) | \(252\) |
\(252~\text{cm}^3\)
| \(6 \times 3\) | \(=\) | \(18\) |
| \(\dfrac{1}{2} \times 6 \times 2\) | \(=\) | \(6\) |
| \(18 + 6\) | \(=\) | \(24\) |
| \(24 \times 10\) | \(=\) | \(240\) |
Cross-section \(24~\text{m}^2\); volume \(240~\text{m}^3\).
| \(8 \times 5 \times 6\) | \(=\) | \(240\) |
| \(240 - 195\) | \(=\) | \(45\) |
| \(45 \div (3 \times 5)\) | \(=\) | \(3\) |
The hole is \(3 \times 5 \times x = 15x\) and \(15x = 45\), so it is \(3\) cm high.
| \(2 + 3 + 2 + 1 + 1 + 1\) | \(=\) | \(10\) |
\(10~\text{cm}^3\) (some cubes are hidden behind the front row)
| \(60 \times 40 \times 25\) | \(=\) | \(60\,000\) |
| \(60 - 5 - 5\) | \(=\) | \(50\) |
| \(40 - 5 - 5\) | \(=\) | \(30\) |
| \(25 - 5\) | \(=\) | \(20\) |
| \(50 \times 30 \times 20\) | \(=\) | \(30\,000\) |
| \(60\,000 - 30\,000\) | \(=\) | \(30\,000\) |
The inside loses two walls in length and width but only the base in height (no lid). Concrete: \(30\,000~\text{cm}^3\).
Common pitfalls
Frequently asked questions
How do you find the volume of a composite solid?
Split it into rectangular prisms and add, or take the missing part away from a whole block.
What is a cross-section?
The face that goes straight through a prism from front to back.
How do you find the volume of a prism with a pentagon face?
Find the area of the pentagon (rectangle plus triangle), then multiply by the length.
How do you find the volume of a block with a hole?
Volume of the whole block minus the volume of the hole.
Do different splits give different answers?
No. Every correct split gives the same volume.