Volumes of Prisms and Cylinders
The volume of any prism or cylinder is the area of its cross-section multiplied by its length: \(V = Ah\), which for a cylinder is \(V = \pi r^2 h\). This NSW Year 9 Core maths subtopic is Stage 5 Core content under NESA outcome MA5‑VOL‑C‑01.
You will find volumes of rectangular, triangular and trapezoidal prisms, cylinders, half-cylinders and quarter-circle prisms, give exact answers in terms of \(\pi\), and subtract a cylindrical hole drilled through a block. You will convert volumes to capacities using \(1~\text{cm}^3 = 1~\text{mL}\), as for a drink can or a water tank, find the volume of a shed with a curved roof, see how changing the radius changes the volume, and work back to a missing dimension. It is part of Year 9 Core Surface Area and Volume.
Theory
Every prism and cylinder follows one rule: volume = area of the cross-section × length. This Year 9 maths page shows how to use it for rectangular and triangular prisms, cylinders, half-cylinders and quadrants. It also covers solids made of two parts, turning a volume into a capacity in litres, and working backwards to a missing length.
A prism is a solid with the same shape all the way through. That shape is the cross-section. A cylinder is like a prism with a circle as its cross-section.
The volume is the amount of space inside a solid. It is measured in cubic units, such as \(\text{cm}^3\) or \(\text{m}^3\).
To find the volume, find the area of the cross-section. Then multiply by the length (or height) of the solid, measured at right angles to the cross-section.
A composite solid is made of two or more simple solids. Add their volumes, or subtract a part that has been cut out, such as a hole.
The capacity of a container is how much it holds. It is measured in millilitres (mL) and litres (L).
For any prism or cylinder, with \(A\) the area of the cross-section and \(h\) the length:
The common cases are the same rule:
| Cross-section | Area \(A\) |
|---|---|
| Triangle | \(\dfrac{1}{2} \times \text{base} \times \text{height}\) |
| Trapezium | \(\dfrac{h}{2}(a + b)\), with \(a\) and \(b\) the parallel sides |
| Circle | \(\pi r^2\) |
| Semicircle | \(\dfrac{1}{2}\pi r^2\) |
| Quadrant | \(\dfrac{1}{4}\pi r^2\) |
How to find the volume of a prism or cylinder
- Find the cross-section: the face that is the same all the way through. If you are given a diameter, halve it to get the radius.
- Work out the area of the cross-section, \(A\).
- Multiply by the length: \(V = A \times h\). For a composite solid, do this for each part, then add (or subtract a hole).
- Finish with cubic units, round as asked, or convert to mL or L. To find a missing length, put the volume in and solve.
Find the area of the triangle first. It is the face that repeats.
| \(A\) | \(=\) | \(\dfrac{1}{2} \times 7 \times 4\) |
| \(\) | \(=\) | \(14 \text{ cm}^2\) |
Multiply by the length of the prism:
| \(V\) | \(=\) | \(14 \times 9\) |
| \(\) | \(=\) | \(\mathbf{126} \textbf{ cm}^\mathbf{3}\) |
The \(9\) cm is not part of the triangle. It is how far the triangle runs.
(i) Find its exact volume.
(ii) Find its volume correct to two decimal places.
(iii) How many millilitres does it hold, to the nearest millilitre?
The radius is half the diameter: \(r = 8 \div 2 = 4\) cm.
(i) The cross-section is a circle, so \(A = \pi r^2\). Leave \(\pi\) in the answer:
| \(V\) | \(=\) | \(\pi r^2 h\) |
| \(\) | \(=\) | \(\pi \times 4^2 \times 9\) |
| \(\) | \(=\) | \(\pi \times 16 \times 9\) |
| \(\) | \(=\) | \(\mathbf{144\pi} \textbf{ cm}^\mathbf{3}\) |
(ii) Now use the \(\pi\) key:
| \(V\) | \(=\) | \(452.389\ldots\) |
| \(\) | \(\approx\) | \(\mathbf{452.39} \textbf{ cm}^\mathbf{3}\) |
(iii) \(1 \text{ cm}^3\) holds \(1\) mL, so the jar holds about 452 mL.
Two solids are joined, so find each one and add. Both are \(30\) cm long.
The box:
| \(V_1\) | \(=\) | \(30 \times 20 \times 25\) |
| \(\) | \(=\) | \(15\,000 \text{ cm}^3\) |
The half-cylinder has radius \(20 \div 2 = 10\) cm. Its cross-section is a semicircle:
| \(A\) | \(=\) | \(\dfrac{1}{2} \times \pi \times 10^2\) |
| \(\) | \(=\) | \(50\pi \text{ cm}^2\) |
| \(V_2\) | \(=\) | \(50\pi \times 30\) |
| \(\) | \(=\) | \(4712.388\ldots \text{ cm}^3\) |
Add the two parts:
| \(V\) | \(=\) | \(15\,000 + 4712.388\ldots\) |
| \(\) | \(\approx\) | \(\mathbf{19\,712.39} \textbf{ cm}^\mathbf{3}\) |
(i) A rectangular prism has a volume of \(1800 \text{ cm}^3\). It is \(20\) cm long and \(9\) cm wide. Find its height.
(ii) A cylinder has radius \(3\) cm and volume \(72\pi \text{ cm}^3\). Find its height.
Write the formula, put in what you know, then solve for the missing length.
(i)
| \(1800\) | \(=\) | \(20 \times 9 \times h\) |
| \(1800\) | \(=\) | \(180h\) |
| \(h\) | \(=\) | \(1800 \div 180\) |
| \(\) | \(=\) | \(\mathbf{10} \textbf{ cm}\) |
(ii)
| \(72\pi\) | \(=\) | \(\pi \times 3^2 \times h\) |
| \(72\pi\) | \(=\) | \(9\pi h\) |
| \(h\) | \(=\) | \(72\pi \div 9\pi\) |
| \(\) | \(=\) | \(\mathbf{8} \textbf{ cm}\) |
Check (ii) forwards: \(\pi \times 9 \times 8 = 72\pi\). The \(\pi\) on both sides cancels.
Common pitfalls
Frequently asked questions
What is the formula for the volume of a prism?
\(V = A \times h\): the area of the cross-section times the length of the prism. The cross-section is the face that is the same all the way through.
How do you find the volume of a cylinder?
Use \(V = \pi r^2 h\). Square the radius, multiply by \(\pi\), then multiply by the height. If you are given the diameter, halve it first.
How do you convert cubic centimetres to millilitres?
\(1 \text{ cm}^3 = 1 \text{ mL}\), so the number stays the same. \(1 \text{ m}^3 = 1000 \text{ L}\).
How do you find the volume of a half-cylinder?
The cross-section is a semicircle, so \(A = \dfrac{1}{2}\pi r^2\). Multiply that area by the length.
What happens to the volume of a cylinder if you triple the radius?
The volume becomes nine times as big, because the radius is squared in \(\pi r^2 h\). \(3^2 = 9\).