Volume of Composite Cylinders
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Theory
A composite solid is made of simpler solids: add the parts, or take the missing piece away. A pipe is a cylinder with a cylinder-shaped hole: \(V = \pi(R^2 - r^2)h\). With \(R = 6\) cm, \(r = 4\) cm and \(h = 10\) cm, \(V = 360\pi - 160\pi = 200\pi~\text{cm}^3\). This Year 8 maths page shows how.
Add when the solid is built from pieces (stacked cylinders, a cylinder on a prism): \(V = V_1 + V_2 + \ldots\). Use \(\pi r^2 h\) for a cylinder and \(lwh\) or \(Ah\) for a prism.
Subtract when something is missing (a hole, a pipe, a ring): \(V = \text{whole} - \text{hole}\). A hole that goes right through is as deep as the solid.
A pipe or tube: \(V = \pi R^2 h - \pi r^2 h = \pi(R^2 - r^2)h\), with outer radius \(R\) and inner radius \(r\). Square each radius first, then subtract.
Find the missing measurement first. Halve every diameter. A wall \(t\) thick gives \(r = R - t\). A cylinder that fits exactly in a square box has a diameter equal to the width of the box.
Keep \(\pi\) to the end: \(80\pi - 45\pi = 35\pi\), rounded on the last line only.
Working backwards: length \(=\) volume \(\div\) ring area. Whole washers from a supply: divide and round down.
Composite cylinders:
How to find a composite volume
- Split the solid into cylinders and prisms, or a whole and a hole.
- Find each radius (halve diameters) and height, in one unit.
- Add or subtract the exact volumes, then round at the end.
| \(V_{\text{large}}\) | \(=\) | \(\pi \times 6^2 \times 10 = 360\pi\) |
| \(V_{\text{hole}}\) | \(=\) | \(\pi \times 4^2 \times 10 = 160\pi\) |
| \(V\) | \(=\) | \(360\pi - 160\pi = 200\pi\) |
\(200\pi~\text{cm}^3\)
| \(V_{\text{small}}\) | \(=\) | \(\pi \times 3^2 \times 4 = 36\pi\) |
| \(V_{\text{large}}\) | \(=\) | \(\pi \times 4^2 \times 4 = 64\pi\) |
| \(V\) | \(=\) | \(36\pi + 64\pi = 100\pi\) |
\(100\pi~\text{cm}^3\)
| \(V_{\text{base}}\) | \(=\) | \(\pi \times 20^2 \times 15 = 6000\pi\) |
| \(V_{\text{post}}\) | \(=\) | \(\pi \times 10^2 \times 90 = 9000\pi\) |
| \(V\) | \(=\) | \(15\,000\pi = 47\,123.8\ldots\) |
\(15\,000\pi~\text{cm}^3\), or \(47\,124~\text{cm}^3\)
| \(R\) | \(=\) | \(6\) |
| \(r\) | \(=\) | \(6 - 1 = 5\) |
| \(V\) | \(=\) | \(\pi \times (36 - 25) \times 200\) |
| \(\) | \(=\) | \(2200\pi = 6911.5\ldots\) |
\(6912~\text{cm}^3\) of steel
| \(V_{\text{block}}\) | \(=\) | \(12 \times 12 \times 5 = 720\) |
| \(V_{\text{hole}}\) | \(=\) | \(\pi \times 6^2 \times 5 = 180\pi\) |
| \(V\) | \(=\) | \(720 - 180\pi = 154.51\ldots\) |
The hole fits exactly, so its diameter is \(12\) cm. \(154.5~\text{cm}^3\) is left.
| \(A\) | \(=\) | \(\pi \times (4 - 3.24) = 0.76\pi\) |
| \(L\) | \(=\) | \(\dfrac{95}{0.76\pi} = 39.78\ldots\) |
About \(40\) cm.
Common pitfalls
Frequently asked questions
How do I find the volume of a pipe?
\(V = \pi(R^2 - r^2)h\): outer cylinder minus inner cylinder.
How do I find the volume of stacked cylinders?
Find each cylinder’s volume and add them.
What is a composite solid?
A solid made from two or more simpler solids, or with a piece taken away.
Why keep pi until the end?
Rounding early makes the final answer less accurate.
How do I find the length of a pipe from its volume?
Divide the volume by the area of the ring.