Volume and Surface Area of Prisms and Cylinders
Learn how to find the volume and surface area of a cylinder and other prisms for IGCSE Maths. This page covers the volume of a prism (cross-section area times length), the volume of a cuboid and cylinder, and how to work out surface area, including the curved surface area of a cylinder. Each method is shown with clear worked examples, with answers given as decimals and in terms of π, followed by auto-marked practice rooms so you can build confidence.
The key idea: stack the slices
Think of a loaf of bread. Every slice is the same shape, and stacking the slices makes the whole loaf. A prism works the same way: it has the same cross-section (one slice) all along its length. So if you can find the area of one slice, the volume is just that area multiplied by how long the stack is.
A cuboid, a triangular prism and a cylinder are all prisms. Each one has a flat slice (its cross-section) at the end: a rectangle, a triangle, or a circle. Find the area of that slice, then multiply by the length.
💡 Volume or surface area?
Volume measures the space inside, so it is in cubic units (cm³) and the formula multiplies three lengths. Surface area is the total area of the outside faces, so it is in square units (cm²) and each part multiplies two lengths. If your answer is a volume it should end in cm³; if it is a surface area it should end in cm².
Volume
For volume, the rule is the same for every prism: find the area of one slice (the cross-section), then multiply by the length.
A cuboid's slice is a rectangle, so its volume is just the three dimensions multiplied together. A cylinder's slice is a circle, so its volume is the circle area \(\pi r^2\) times the height \(h\). A triangular prism's slice is a triangle, so find \(\tfrac{1}{2}\times\text{base}\times\text{height}\) and multiply by the length.
Surface area
Surface area is a different skill from volume. It is the total area of all the outside faces added together. The clearest way to see every face is to open the solid out flat into its net, work out the area of each piece, then add them all up.
Cuboid: add the six faces
Start with the cuboid and its three dimensions. This one measures \(5\) by \(4\) by \(3\):
Now open it out flat into its net: six rectangles, with opposite faces matching, so there are three pairs. The dimensions sit on the outside edges and the area of each face is written in the middle:
The shortcut is to add the three different faces and double, since each appears twice: \(2(20 + 15 + 12) = 2 \times 47 = 94 \text{ cm}^2\). Same answer, less writing.
Triangular prism: two triangles and three rectangles
The net of a triangular prism opens into two triangles (the matching ends) and three rectangles (the sides). Each rectangle has one side equal to the length of the prism. Here is the net for a prism with a \(3, 4, 5\) right-angled triangle and length \(10\):
Step 1: the two triangles each have area \(\tfrac{1}{2}\times 3 \times 4 = 6\), giving \(12\) altogether.
Step 2: the three rectangles are \(3\times 10 = 30\), \(4\times 10 = 40\) and \(5\times 10 = 50\), giving \(120\).
Step 3: add everything: \(12 + 120 = 132 \text{ cm}^2\).
Cylinder: unroll the curved surface
The net of a cylinder is two circles (the ends) plus one rectangle (the curved surface). The key link is that the rectangle's width is exactly the circumference of the circle, \(2\pi r\), and its height is the height of the cylinder, \(h\). Press play (or drag the slider) to watch the net fold itself into the cylinder: the rectangle rolls into the tube and the two circles close the ends.
So the curved surface area is \(2\pi r \times h\), and adding the two circular ends (each \(\pi r^2\)) gives the total.
Worked examples
💡 Example 1: cuboid volume
A cuboid measures \(8\) cm by \(4\) cm by \(5\) cm. Find its volume.
What's happening?
Multiply the three dimensions together.
Volume is in cubic units, so the answer is in cm³.
💡 Example 2: cuboid surface area
Find the total surface area of the same cuboid (\(8\) by \(4\) by \(5\) cm).
What's happening?
There are three pairs of matching faces.
Find the area of each different face, add them, then double.
💡 Example 3: cylinder volume
A cylinder has radius \(3\) cm and height \(10\) cm. Find its volume, in terms of \(\pi\) and to 3 significant figures.
What's happening?
Square the radius first: \(3^2 = 9\), then multiply by the height.
Leave it as \(90\pi\) for an exact answer, or multiply out for a decimal.
💡 Example 4: cylinder surface area
A closed cylinder has radius \(5\) cm and height \(12\) cm. Find its total surface area, to 3 significant figures.
What's happening?
The curved part is \(2\pi r h = 120\pi\).
The two circular ends add \(2\pi r^2 = 50\pi\). Add them for the total.
💡 Example 5: triangular prism
A prism has a right-angled triangular cross-section with base \(6\) cm and height \(4\) cm, and a length of \(15\) cm. Find its volume.
What's happening?
First find the area of the triangular cross-section.
Then multiply by the length of the prism. This works for any prism: find the cross-section area, then multiply by the length.
🔑 Key points
- Volume of any prism \(= \text{cross-section area} \times \text{length}\).
- Cuboid: \(V = w\,d\,h\); cylinder: \(V = \pi r^2 h\).
- Curved surface area of a cylinder \(= 2\pi r h\).
- Total surface area of a closed cylinder \(= 2\pi r h + 2\pi r^2\).
- Volume is in cm³; surface area is in cm².
- "In terms of \(\pi\)" means leave \(\pi\) in the answer.
⚠️ Common pitfalls
- Using the diameter instead of the radius in \(\pi r^2 h\).
- Forgetting the two ends when a cylinder is closed.
- Mixing up volume (cubic units) and surface area (square units).
- Squaring \(\pi r\) instead of just \(r\).
- Forgetting to find the cross-section area first for a prism.
- Rounding too early; keep full accuracy until the final line.