Maths
What is surface area?
Quick answer
Surface area is the total area of all the outside faces of a 3D shape. For a cube with side a, it is 6a², because a cube has six equal square faces.
In one sentence
Surface area is the total area of all the faces that cover the outside of a 3D shape.
Explained
Imagine wrapping a present. The amount of wrapping paper you need, with nothing wasted and no overlap, is the surface area of the box. For paint on a wall or the leather on a football, the same idea applies.
box
ruler
pencil
To find the surface area of a shape, work out the area of each face and add them together. A good way is to draw the net, the shape unfolded flat. For a cube with side 3 cm, the net is six squares. One square is 3 × 3 = 9 cm², so the total is 6 × 9 = 54 cm². In general, a cube with side a has surface area 6a².
A cuboid (a box) has three pairs of matching rectangles: top and bottom, front and back, left and right. If the length is l, width w and height h, then
Surface area = 2lw + 2lh + 2wh, or 2(lw + lh + wh).
For a box 5 cm long, 3 cm wide and 2 cm high: lw = 15, lh = 10 and wh = 6. The total is 2 × (15 + 10 + 6) = 2 × 31 = 62 cm².
A cylinder has two circles and a curved side that opens out into a rectangle: SA = 2πr² + 2πrh. With r = 2 cm and h = 5 cm, that is about 8π + 20π = 28π, around 88 cm².
Surface area is measured in square units (cm², m²). Do not mix it up with volume, which measures the space inside in cubic units. Surface area relies on area and on knowing the faces of 3D shapes.
Examples
| Shape | Measurements | Surface area |
|---|---|---|
| Cube | side 3 cm | 54 cm² |
| Cube | side 5 cm | 150 cm² |
| Cuboid | 5 by 3 by 2 cm | 62 cm² |
| Cylinder | r = 2 cm, h = 5 cm | about 88 cm² |
Common mistakes
- Forgetting the hidden faces. The bottom and back count too.
- Mixing up volume and surface area. Surface area is the covering; volume is the filling.
- Using cm instead of cm². The answer is an area.
Quick quiz
Surface area
Check what you know with 4 quick questions.
Question 1 of 4
What is the surface area of a cube with sides 3 cm?
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Related topics
3D shapes
3D shapes are solid shapes with length, width and height, like cubes, spheres, cylinders and pyramids. They have faces, edges and vertices.
Area
Area is the amount of flat space inside a shape, measured in square units like cm² or m². A rectangle 4 cm by 3 cm has an area of 4 × 3 = 12 cm².
Volume
Volume is the amount of space a 3D object takes up, measured in cubic units like cm³ or m³. A box 4 cm by 3 cm by 2 cm has a volume of 24 cm³.
Area of a circle
The area of a circle is pi times the radius squared: A = πr². A circle with radius 5 cm has an area of about 78.5 square centimetres.
Area of a triangle
The area of a triangle is half of its base times its perpendicular height: A = ½ × b × h. A triangle with base 8 cm and height 5 cm has an area of 20 square centimetres.
Pi (π)
Pi (π) is the number you get when you divide any circle's circumference by its diameter. It is about 3.14, and its digits go on forever without repeating.