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Surface Area of a Prism

FoundationHigherAQAEdexcelOCR

This free Foundation and Higher GCSE Maths worksheet on surface area of a prism helps you revise surface area of prisms. Questions build from straightforward to exam standard, with full worked answers in the mark scheme — ideal for revision or homework. Suitable for AQA, Edexcel and OCR. Sketch the net so you don't miss a face, then add the areas.

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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.

Foundation

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Higher

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Topic overview

The surface area of a prism is the two identical end faces plus the rectangular faces joining them.

The efficient method uses the fact that the curved or side surface unrolls into one large rectangle. Its area is the perimeter of the end face multiplied by the length of the prism.

So the total is \(2 \times \text{area of end} + \text{perimeter of end} \times \text{length}\). This works for any prism, whatever the shape of its cross-section, which makes it far more useful than memorising separate formulae.

Revision notes

The general method

Surface area is \(2 \times \text{end area} + \text{end perimeter} \times \text{length}\).

For a triangular prism with end area \(6\)cm², end perimeter \(12\)cm and length \(10\)cm: \(2 \times 6 + 12 \times 10 = 132\)cm².

Why the perimeter works

Unrolling the sides of a prism gives one rectangle whose height is the prism's length and whose width is the perimeter of the end.

This is why the method works for any cross-section, including irregular ones.

Cylinders

A cylinder is a prism with a circular cross-section, so the same logic applies.

The curved surface is \(2\pi r h\) — the circumference times the height — and the two circular ends add \(2\pi r^2\), giving \(2\pi r h + 2\pi r^2\) in total.

Key points

  • A prism has two identical ends joined by rectangles.
  • Surface area is \(2 \times\) end area \(+\) end perimeter \(\times\) length.
  • The sides unroll into one rectangle.
  • The method works for any cross-section.
  • A cylinder's curved surface is \(2\pi r h\).
  • Surface area uses squared units.

Worked examples

Example 1

A prism has end area \(15\)cm², end perimeter \(18\)cm and length \(8\)cm. Find its surface area.

Working

\[2 \times 15 = 30\]the two identical ends
\[18 \times 8 = 144\]the perimeter times the length gives the sides
\[30 + 144 = 174\text{cm}^2\]add for the total

Example 2

Find the curved surface area of a cylinder with radius \(3\)cm and height \(10\)cm, in terms of \(\pi\).

Working

\[2\pi r h = 2\pi \times 3 \times 10\]use the curved surface formula
\[= 60\pi\text{cm}^2\]state the area

Example 3

Find the total surface area of that cylinder, in terms of \(\pi\).

Working

\[60\pi\]the curved surface
\[2\pi r^2 = 2\pi \times 9 = 18\pi\]the two circular ends
\[= 78\pi\text{cm}^2\]add for the total

Common mistakes

  • Forgetting the two end faces.

    The curved or side surface is only part of the total.

  • Using the area of the end instead of its perimeter for the sides.

    The sides unroll to a rectangle whose width is the perimeter.

  • Using the diameter in the cylinder formulae.

    Both 2πrh and 2πr² need the radius.

  • Confusing curved surface with total surface area.

    Read carefully which the question wants.

Exam tips

  • Use end area and end perimeter — it works for every prism.
  • Check whether the question wants curved or total surface area.
  • Halve the diameter before using cylinder formulae.
  • Give answers in terms of π when exact values are wanted.

Key terms

Prism
A solid with a constant cross-section.
Cross-section
The shape of the identical end faces.
Curved surface area
The area of the side of a cylinder, excluding the ends.
Perimeter
The distance around the end face.

Related topics

Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.