Volume of Prisms and Cylinders
Theory
The volume of a prism is the area of its cross-section times its length, \(V = Ah\), and a cylinder gives \(V = \pi r^{2}h\). This Year 10A maths page revises Year 9: triangular, trapezium, sector and composite prisms, capacity in litres and kilolitres, pouring water between containers, and working backwards to a radius or height.
Volume is the amount of space inside a solid, measured in cubic units such as \(\text{cm}^3\) or \(\text{m}^3\). Capacity is how much a container holds, measured in mL, L or kL.
A prism has the same cross-section all the way through. Slice it anywhere parallel to the end and you get the same shape with the same area \(A\). So its volume is \(V = Ah\), where \(h\) is the length (or height) of the prism, measured at right angles to the cross-section.
A cylinder is a prism with a circular cross-section of area \(\pi r^{2}\), so \(V = \pi r^{2}h\).
The cross-section can be any shape you can find the area of: a triangle, a trapezium, an L-shape made of rectangles, a semicircle or a sector.
| Solid | Cross-section area \(A\) | Volume |
|---|---|---|
| Rectangular prism | \(lw\) | \(V = lwh\) |
| Triangular prism | \(\dfrac{1}{2}bh_{\triangle}\) | \(V = \dfrac{1}{2}bh_{\triangle} \times \text{length}\) |
| Trapezium prism | \(\dfrac{1}{2}(a + b)h_{\text{trap}}\) | \(V = A \times \text{length}\) |
| Cylinder | \(\pi r^{2}\) | \(V = \pi r^{2}h\) |
| Half cylinder | \(\dfrac{1}{2}\pi r^{2}\) | \(V = \dfrac{1}{2}\pi r^{2}h\) |
| Sector prism (wedge) | \(\dfrac{\theta}{360}\pi r^{2}\) | \(V = \dfrac{\theta}{360}\pi r^{2} \times \text{length}\) |
How to find the volume of a prism or cylinder
- Identify the cross-section: the face that is the same all the way through.
- Find its area \(A\). Split a composite shape into rectangles and triangles; halve a diameter for a circle.
- Multiply by the length of the prism, with every length in the same unit.
- Convert to mL, L or kL if the question asks for capacity.
Problems
- Partly full: use the depth of the water, not the height of the container. If it is a percentage full, find the full volume and take that percentage of it.
- Pouring: the volume stays the same. Set it equal to the new container's base area times the unknown depth.
- Working backwards: substitute and solve. If the sides of the cross-section are \(x\) and \(2x\), the area is an expression in \(x^{2}\): solve for \(x^{2}\), then take the square root. For a radius, divide by \(\pi h\) and take the square root; for a cube or a prism with sides \(x\), \(2x\), \(3x\), take a cube root.
- How many full ones? Divide and round down: a part glass or part day does not count.
- Cost: multiply the volume by the price per cubic metre (concrete, soil, water). For example, \(2.5\text{ m}^3\) of soil at \(\$48\) per cubic metre costs \(2.5 \times 48 = \$120\).
Find the area of the cross-section, then multiply by the length.
| \(A\) | \(=\) | \(\dfrac{1}{2}(6 + 10) \times 4 = 32\text{ cm}^2\) |
| \(V\) | \(=\) | \(Ah\) |
| \(\) | \(=\) | \(32 \times 15\) |
| \(\) | \(=\) | \(\mathbf{480}\text{ cm}^3\) |
(i) its exact volume
(ii) its capacity in litres, correct to two decimal places.
Halve the diameter: \(r = 7\) cm.
(i)
| \(V\) | \(=\) | \(\pi r^{2}h\) |
| \(\) | \(=\) | \(\pi \times 7^{2} \times 20\) |
| \(\) | \(=\) | \(\mathbf{980\pi}\text{ cm}^3\) |
(ii) \(1\text{ cm}^3 = 1\text{ mL}\) and \(1000\text{ mL} = 1\text{ L}\).
| \(980\pi\) | \(=\) | \(3078.76\ldots\text{ mL}\) |
| \(\) | \(\approx\) | \(\mathbf{3.08}\text{ L}\) |
The volume of water stays the same. Find it, then divide by the base area of the tub.
| \(V\) | \(=\) | \(\pi \times 15^{2} \times 20 = 4500\pi\text{ cm}^3\) |
| \(50 \times 30 \times d\) | \(=\) | \(4500\pi\) |
| \(d\) | \(=\) | \(\dfrac{4500\pi}{1500} = 3\pi\) |
| \(\) | \(=\) | \(9.4247\ldots\) |
| \(\) | \(\approx\) | \(\mathbf{9.42}\text{ cm}\) |
Change the capacity to cubic metres so it matches the height in metres: \(3000\text{ L} = 3\text{ m}^3\).
| \(\pi r^{2} \times 1.6\) | \(=\) | \(3\) |
| \(r^{2}\) | \(=\) | \(\dfrac{3}{1.6\pi}\) |
| \(r\) | \(=\) | \(\sqrt{\dfrac{3}{1.6\pi}}\) |
| \(\) | \(=\) | \(0.7725\ldots\) |
| \(\) | \(\approx\) | \(\mathbf{0.77}\text{ m}\) |
Check: \(\pi \times 0.7725^{2} \times 1.6 \approx 3.00\text{ m}^3\).
Common pitfalls
Frequently asked questions
What is the formula for the volume of a prism?
The volume of any prism is the area of its cross-section multiplied by its length: \(V = Ah\).
How do you find the volume of a cylinder?
Square the radius, multiply by \(\pi\), then multiply by the height: \(V = \pi r^{2}h\). If you are given the diameter, halve it first.
How many litres are in a cubic metre?
One cubic metre holds \(1000\) litres (one kilolitre). One cubic centimetre holds one millilitre.
How do you find the height of a cylinder from its volume?
Substitute into \(V = \pi r^{2}h\), then divide: \(h = \dfrac{V}{\pi r^{2}}\).
What is the difference between volume and capacity?
Volume is the space a solid takes up, in cubic units. Capacity is how much a container can hold, in mL, L or kL. They are linked by \(1\text{ cm}^3 = 1\text{ mL}\).