Surface Area of Prisms and Pyramids
Theory
The surface area of a prism is the total area of all its faces. This Year 9 maths page uses nets to find the surface area of cubes, rectangular prisms and triangular prisms, extends it to any right prism, handles open boxes and missing edges, and works backwards from a surface area to an edge length.
A prism is a solid with two identical, parallel ends joined by flat faces. The shape of the ends is the cross-section. In a right prism, the kind on this page, those faces are rectangles that meet the ends at right angles.
The surface area (SA) is the total area of all the faces. It is measured in square units, such as \(\text{cm}^2\).
A net is the solid unfolded flat. Drawing it shows every face, so none is missed.
A cube has \(6\) faces, a rectangular prism \(6\) and a triangular prism \(5\): \(2\) triangles and \(3\) rectangles. An open solid is missing a face, such as the lid of a box.
Any right prism, including a triangular, trapezium or L-shaped prism:
How to find the surface area of a prism
- Sketch the net, or list the faces: two ends and one rectangle for each side of the end.
- Find any missing lengths, such as a sloping edge, with Pythagoras.
- Work out the area of each face. Matching faces can be doubled.
- Leave out any face that is not there, such as a missing lid.
- Add the areas and give the answer in square units.
The six faces come in three matching pairs. Work out one of each pair.
| \(9 \times 6\) | \(=\) | \(54\) |
| \(9 \times 4\) | \(=\) | \(36\) |
| \(6 \times 4\) | \(=\) | \(24\) |
Add them, then double.
| \(SA\) | \(=\) | \(2 \times (54 + 36 + 24)\) |
| \(\) | \(=\) | \(2 \times 114 = \mathbf{228} \text{ cm}^2\) |
With no lid there are five faces, not six. List the ones that are there.
| \(\text{base}\) | \(=\) | \(15 \times 10 = 150\) |
| \(\text{2 long sides}\) | \(=\) | \(2 \times 15 \times 8 = 240\) |
| \(\text{2 ends}\) | \(=\) | \(2 \times 10 \times 8 = 160\) |
| \(SA\) | \(=\) | \(150 + 240 + 160\) |
| \(\) | \(=\) | \(\mathbf{550} \text{ cm}^2\) |
A closed box would add another \(150 \text{ cm}^2\) for the lid.
The sloping side of the triangle is not given. Find it with Pythagoras.
| \(c^2\) | \(=\) | \(9^2 + 12^2 = 225\) |
| \(c\) | \(=\) | \(15 \text{ cm}\) |
Two triangular ends, then three rectangles, each \(7\) cm long.
| \(\text{ends}\) | \(=\) | \(2 \times \dfrac{1}{2} \times 9 \times 12 = 108\) |
| \(\text{rectangles}\) | \(=\) | \((9 + 12 + 15) \times 7 = 252\) |
| \(SA\) | \(=\) | \(108 + 252 = \mathbf{360} \text{ cm}^2\) |
(i) Find the surface area of a cube with edges of \(7\) cm.
(ii) Another cube has a surface area of \(384 \text{ cm}^2\). How long is each edge?
(i) A cube has six identical square faces.
| \(SA\) | \(=\) | \(6s^2 = 6 \times 7^2\) |
| \(\) | \(=\) | \(6 \times 49 = \mathbf{294} \text{ cm}^2\) |
(ii) Undo the steps: divide by \(6\) to get one face, then take the square root.
| \(6s^2\) | \(=\) | \(384\) |
| \(s^2\) | \(=\) | \(64\) |
| \(s\) | \(=\) | \(\sqrt{64} = \mathbf{8} \text{ cm}\) |
Common pitfalls
Frequently asked questions
What is the surface area of a prism?
It is the total area of all the faces of the prism: the two ends plus the rectangles that join them.
How do you find the surface area of a rectangular prism?
Find the three different faces, add them, then double: \(SA = 2(lb + lh + bh)\). A \(9\) by \(6\) by \(4\) cm prism has \(SA = 228 \text{ cm}^2\).
How do you find the surface area of a triangular prism?
Add the two triangular ends to the three rectangles. Each rectangle is as long as the prism and as wide as one side of the triangle.
How do you find the surface area of any right prism?
Add the two ends to the side faces. All the side faces together make a rectangle as long as the perimeter of the end and as wide as the prism is long: \(SA = 2 \times \text{end} + \text{perimeter} \times \text{length}\).
What is a net of a prism?
A net is the prism unfolded into a flat shape. It shows every face at once, which makes it easy to add up the surface area.
How do you find the side of a cube from its surface area?
Divide the surface area by \(6\) to get one face, then take the square root. \(384 \div 6 = 64\), so the edge is \(8\) cm.