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Surface Area of Prisms and Cylinders

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How to work out the surface area of cubes, cuboids, triangular prisms and cylinders by adding the areas of their faces, and how to use it in problems about paint, labels and boxes without lids.

Surface area in the exam

The surface area of a 3D shape is the total area of all its faces. Think of it as the amount of paper you would need to cover the shape completely, with no gaps and no overlaps.

In the exam you may be asked to work out the total surface area of a cuboid or a cube from a diagram, for 2 or 3 marks. Some questions link it with volume: the surface area of a cube from its volume, or its volume from its surface area.

Later in a calculator paper, a problem for 4 or 5 marks may ask how many tins of paint you need to cover some cylinders, or whether there is enough paint.

There is no single formula to learn for every shape. You work out the area of each face and add them up. Surface area is an area, so it is measured in square units such as \text{cm}^2 or \text{m}^2

Faces of a cuboid

The surface area of a 3D shape is the total area of all its faces. A cuboid has six faces, in three matching pairs: the top and the base, the front and the back, and the two ends.

The six faces come in three matching pairs: - top and base: 4\times3=12\text{ cm}^2 each - front and back: 4\times2=8\text{ cm}^2 each - two ends: 3\times2=6\text{ cm}^2 each

Add all six: 12+12+8+8+6+6=52. The surface area of the cuboid is 52\text{ cm}^2

Cuboids and cubes

You don't need to work out all six faces one by one. A cuboid always has six faces in three pairs, so work out three different faces, add them, then double:

\text{surface area}=2\times(\text{top}+\text{front}+\text{end})

Each face uses two of the three lengths. The top uses the length and the width, the front uses the length and the height, and the end uses the width and the height.

A cube has six faces that are all the same square. So the surface area of a cube is 6 times the area of one face. A cube with edges 4 cm long has surface area 6\times4\times4=96\text{ cm}^2

Worked example 1: The surface area of a cuboid

A cuboid is 7 cm long, 4 cm wide and 2 cm high. Work out the total surface area of the cuboid.

  1. Label the three faces you can see: the top, the front and one end.
  2. Top: 7\times4=28\text{ cm}^2. Front: 7\times2=14\text{ cm}^2
  3. End: 4\times2=8\text{ cm}^2. One of each face adds up to 28+14+8=50\text{ cm}^2

  4. Your turn. There are two of each face. Work out the total surface area, in \text{cm}^2

    2\times50=100\text{ cm}^2

  5. Answer: 100\text{ cm}^2

Worked example 2: From surface area to volume

Sam covers every face of a wooden cube with paper, with no gaps and no overlaps. He uses exactly 486\text{ cm}^2 of paper.

Work out the volume of the cube.

  1. The paper covers the surface area, 486\text{ cm}^2. A cube has 6 faces, and they are all the same square.

  2. Your turn. Work out the area of one face, in \text{cm}^2

    486\div6=81\text{ cm}^2

  3. Each face is a square of area 81\text{ cm}^2, so each edge is \sqrt{81}=9 cm, because 9\times9=81
  4. Volume =9\times9\times9=81\times9=729
  5. Answer: 729\text{ cm}^3

Triangular prisms and other prisms

A prism has the same cross-section all the way through. A triangular prism has 5 faces: two triangles, one at each end, and three rectangles joining them.

The two triangles are the same, so work out one and double it, using \frac12\times\text{base}\times\text{height}. The height must be at right angles to the base.

Each rectangle is one side of the triangle multiplied by the length of the prism. The three rectangles are usually different sizes, so work out all three.

The same idea works for any prism, even an L shape. Add the two ends, then add one rectangle for every side of the cross-section, each as long as the prism.

Worked example 3: A triangular prism

A prism is 9 cm long. Its ends are right-angled triangles with sides 5 cm, 12 cm and 13 cm.

Work out the total surface area of the prism.

  1. The prism has two triangles and three rectangles. The 5 cm and 12 cm sides meet at the right angle.

  2. Your turn. Work out the area of one triangle, in \text{cm}^2

    \frac12\times12\times5=30\text{ cm}^2, so the two triangles are 60\text{ cm}^2

  3. The rectangles: 5\times9=45, 12\times9=108 and 13\times9=117
  4. 45+108+117=270\text{ cm}^2
  5. Total: 60+270=330\text{ cm}^2
  6. Answer: 330\text{ cm}^2

Cylinders

A cylinder is a prism with a circle as its cross-section, like a tin of beans. Its surface is two circles, the top and the bottom, and one curved surface.

Peel the label off a tin and lay it flat: it is a rectangle. One side of the rectangle is the height of the tin. The other side went once round the circle, so it is the circumference, 2\pi r

So, where r is the radius and h is the height:

\text{curved surface area}=2\pi rh

\text{area of each end}=\pi r^2

For the total surface area of a closed cylinder, add the curved surface and both ends. If you are given the diameter, halve it to get the radius first.

Worked example 4: A cylinder in terms of pi

A closed cylinder has radius 3 cm and height 8 cm.

Work out the total surface area of the cylinder. Give your answer in terms of \pi

  1. The circumference is 2\times\pi\times3=6\pi cm.
  2. Curved surface area: 6\pi\times8=48\pi\text{ cm}^2

  3. Your turn. Work out the area of one circular end. Give your answer in terms of \pi. You can type pi.

    Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

    \pi\times3^2=9\pi\text{ cm}^2, so the two ends are 18\pi\text{ cm}^2

  4. Total: 48\pi+18\pi=66\pi
  5. Answer: 66\pi\text{ cm}^2

Open boxes, lids and labels

Read the question carefully to see which faces are really there. A box with no lid has only 5 faces. A plant pot with a base but no top has one circle, not two.

A label on a tin, or a tube open at both ends, is the curved surface only. If a question says "paint completely" or "with a top and a bottom", include every face.

On a calculator paper, use the \pi button and keep the full numbers in your calculator. Round only the final answer.

Keep all the lengths in the same unit. Large surfaces are often in \text{m}^2, and 1\text{ m}^2=100\times100=10\,000\text{ cm}^2

Worked example 5: A cylinder with no lid

A cake tin is a cylinder with a base but no lid. Its diameter is 20 cm and its height is 8 cm.

Work out the area of metal needed to make the tin. Give your answer correct to 1 decimal place.

  1. The tin has a curved surface and one circle, the base. There is no lid.

  2. Your turn. What is the radius of the tin, in cm?

    The radius is half the diameter: 20\div2=10 cm.

  3. Base: \pi\times10^2=314.159\ldots\text{ cm}^2
  4. Curved surface: the circumference is \pi\times20=62.83\ldots cm, so the area is 62.83\ldots\times8=502.654\ldots\text{ cm}^2

  5. Your turn. Add the base and the curved surface. Give your answer correct to 1 decimal place.

    314.159\ldots+502.654\ldots=816.81\ldots, which is 816.8 to 1 decimal place.

  6. Answer: 816.8\text{ cm}^2

Worked example 6: How many tins of paint

Ravi is going to paint the curved surface of 4 round pillars. He will not paint the tops or the bottoms.

Each pillar is a cylinder with diameter 0.5 m and height 2.4 m. One tin of paint covers 6\text{ m}^2

How many tins of paint does Ravi need?

  1. Only the curved surfaces are painted. The circumference of one pillar is \pi\times0.5=1.570\ldots m.

  2. Your turn. The curved surface unrolls into a rectangle 1.570\ldots m long and 2.4 m high. Work out its area, correct to 2 decimal places.

    1.570\ldots\times2.4=3.769\ldots\text{ m}^2, which is 3.77\text{ m}^2 to 2 decimal places.

  3. Four pillars: 4\times3.769\ldots=15.079\ldots\text{ m}^2
  4. Number of tins: 15.079\ldots\div6=2.51\ldots

  5. Your turn. Ravi can only buy whole tins. How many tins does he need?

    2 tins cover only 12\text{ m}^2, which is not enough, so he needs 3 tins.

  6. Answer: 3 tins

Common mistakes

  • Working out the volume instead. Surface area adds up the areas of the faces, in square units.
  • Getting the pairs of faces wrong. A cuboid has three different pairs of faces, not four faces the same.
  • Treating the curved surface as diameter \times height. Its length is the circumference, 2\pi r or \pi d
  • Using the diameter as the radius. Halve the diameter before using \pi r^2 or 2\pi r
  • Leaving out a face, or adding one that isn't there. Check for lids, bases and open ends.
  • Dividing a cube's surface area by 4. A cube has 6 equal faces, so divide by 6

Practice questions

More on this topic: Surface Area of Prisms and Cylinders worksheet with full solutions