Composite Shapes for Volume
A composite solid is still a prism when the same face runs all the way through. In NSW Year 8 maths (NESA outcome MA4‑VOL‑C‑01) the work goes into the area of that cross-section, built by adding simple shapes or taking a hole away, before using \(V = Ah\).
You will find missing lengths first, split L-shapes, plus signs and I-beams into rectangles, add a triangle or trapezium for a roof or a ramp, and use Pythagoras' theorem for a sloping height. You will also find the concrete in a culvert or a hollow building block, the litres in a pond with a shelf, and how many bags of soil fill a garden bed. It is part of Year 8 Volume.
Theory
A composite solid is still a prism when the same face runs all the way through. Build the area of that face from shapes you know (add the parts, or take a hole away), then multiply by the height: \(V = Ah\). This Year 8 maths page shows how.
\(V = Ah\) has not changed; all the work is in finding \(A\).
Add the parts. One straight line splits the cross-section into known shapes: \(A = A_1 + A_2 + \ldots\)
Take the hole away. For a cut-out, groove or hole right through: \(A = A_{\text{outside}} - A_{\text{hole}}\).
Missing lengths first: total minus the known part. Equal squares: count them and multiply by one square’s area.
Sloping top? If only slant lengths are given, use Pythagoras’ theorem to find the perpendicular height first.
Answer the question asked. Cost \(=\) volume \(\times\) price per cubic metre; \(1~\text{m}^3 = 1000\) L. Bags of soil: divide and round up. Whole objects from a supply: divide and round down; the remainder is left over.
Working backwards. \(A = V \div h\), then put the unknown into the area and solve.
Composite prisms:
| Cross-section | Area |
|---|---|
| parts added | \(A = A_1 + A_2\) |
| hole taken away | \(A = A_{\text{outside}} - A_{\text{hole}}\) |
| then | \(V = Ah\) |
How to find a composite volume
- Split the cross-section into known shapes, or see it as a shape with a hole.
- Find any missing lengths, then the area \(A\).
- Multiply by \(h\) and answer the question asked.
Split the L into the \(4\) cm by \(2\) cm bottom strip and a \(2\) cm by \(3\) cm piece above it on the left (\(5 - 2 = 3\)).
| \(A\) | \(=\) | \(4 \times 2 + 2 \times 3\) |
| \(\) | \(=\) | \(8 + 6 = 14\) |
| \(V\) | \(=\) | \(14 \times 3 = 42\) |
\(42~\text{cm}^3\)
| \(V_{\text{block}}\) | \(=\) | \(6 \times 6 \times 8 = 288\) |
| \(V_{\text{hole}}\) | \(=\) | \(\dfrac{1}{2} \times 4 \times 4 \times 8 = 64\) |
| \(V\) | \(=\) | \(288 - 64 = 224\) |
\(224~\text{cm}^3\)
(i) the area of the cross-section
(ii) the volume
| \(A\) | \(=\) | \(10 \times 3 + 4 \times 6 = 54\) |
| \(V\) | \(=\) | \(54 \times 5 = 270\) |
(i) \(54~\text{cm}^2\) (the upright is \(9 - 3 = 6\) cm tall)
(ii) \(270~\text{cm}^3\)
(i) the volume of steel
(ii) how many whole blocks come from \(0.5~\text{m}^3\) of steel, and how much is left
| \(A\) | \(=\) | \(30 \times 20 - 10 \times 8\) |
| \(\) | \(=\) | \(520\) |
| \(V\) | \(=\) | \(520 \times 25\) |
| \(\) | \(=\) | \(13\,000\) |
| \(n\) | \(=\) | \(\dfrac{500\,000}{13\,000}\) |
| \(\) | \(=\) | \(38.46\ldots\) |
(i) \(13\,000~\text{cm}^3\)
(ii) \(0.5~\text{m}^3 = 500\,000~\text{cm}^3\): \(38\) whole blocks, with \(6000~\text{cm}^3\) left.
(i) the height \(h\) of the lid
(ii) the volume
| \(h^2\) | \(=\) | \(1.0^2 - 0.8^2 = 0.36\) |
| \(h\) | \(=\) | \(0.6\) |
| \(A\) | \(=\) | \(1.76 + 0.48 = 2.24\) |
| \(V\) | \(=\) | \(2.24 \times 2.5 = 5.6\) |
(i) \(0.6\) m
(ii) \(5.6~\text{m}^3\)
| \(A\) | \(=\) | \(3060 \div 15 = 204\) |
| \(240 - x^2\) | \(=\) | \(204\) |
| \(x^2\) | \(=\) | \(36\) |
| \(x\) | \(=\) | \(6\) |
The hole has sides of \(6\) cm.
Common pitfalls
Frequently asked questions
What is a composite prism?
A prism whose cross-section is made of two or more simple shapes.
How do I find its volume?
Find the area of the cross-section by adding or subtracting parts, then multiply by the height.
What if there is a hole?
Subtract the hole’s area from the cross-section before multiplying.
How many bags do I need?
Divide the volume by one bag and round up.
How many whole blocks can I make?
Divide the supply by one block’s volume and round down.