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Similar Shapes

FoundationHigherAQAEdexcelOCR

Master similar shapes for GCSE Maths with structured, exam-style practice. This Foundation and Higher resource covers using scale factors with similar shapes and includes a complete mark scheme showing the steps examiners reward. Suitable for AQA, Edexcel and OCR. Free to download as a PDF. Corresponding sides of similar shapes are in the same ratio.

Free downloads

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

WorksheetFoundation · PDFMark SchemeFoundation · PDF

Higher

WorksheetHigher · PDFMark SchemeHigher · PDF

Topic overview

Two shapes are similar when one is an enlargement of the other. Corresponding angles are equal and corresponding sides are in the same ratio.

That ratio is the scale factor, found by dividing a length on one shape by the matching length on the other. Once known, it gives every other missing length.

Area and volume scale differently. If lengths scale by \(k\), areas scale by \(k^2\) and volumes by \(k^3\). Applying the linear scale factor to an area is one of the most common and costly errors in the topic.

Revision notes

Finding the scale factor

Divide a length on the larger shape by the corresponding length on the smaller.

If matching sides are \(6\)cm and \(9\)cm, the scale factor is \(1.5\). Make sure the lengths genuinely correspond by checking the angles.

Finding missing lengths

Multiply by the scale factor to go from small to large, or divide to go the other way.

With a scale factor of \(1.5\), a side of \(4\)cm on the small shape matches \(6\)cm on the large one.

Area and volume scale factors

Areas scale by \(k^2\) and volumes by \(k^3\).

So if lengths double, the area is four times larger and the volume eight times larger. This surprises students but follows directly from multiplying two or three dimensions.

Key points

  • Similar shapes have equal angles.
  • Corresponding sides are in the same ratio.
  • The scale factor is that ratio.
  • Multiply by \(k\) to enlarge, divide to reduce.
  • Areas scale by \(k^2\).
  • Volumes scale by \(k^3\).

Worked examples

Example 1

Two similar triangles have matching sides \(4\)cm and \(10\)cm. Find the scale factor.

Working

\[\frac{10}{4}\]divide the larger by the smaller
\[= 2.5\]state the scale factor

Example 2

Using that scale factor, find the image of a \(6\)cm side.

Working

\[6 \times 2.5\]multiply by the scale factor
\[= 15\text{cm}\]state the length

Example 3

Two similar shapes have a length scale factor of \(3\). How do their areas compare?

Working

\[k^2 = 3^2\]areas scale by the square of the scale factor
\[= 9 \text{ times larger}\]state the area ratio

Common mistakes

  • Using the linear scale factor for area.

    Areas scale by k², so doubling the lengths quadruples the area.

  • Matching the wrong pair of sides.

    Check the angles to confirm which sides correspond.

  • Assuming similar means congruent.

    Similar shapes have the same angles but usually different sizes.

  • Using k³ for area.

    Cubing applies to volume, squaring to area.

Exam tips

  • Check corresponding angles before pairing sides.
  • Write the scale factor down before finding lengths.
  • Square it for areas and cube it for volumes.
  • Sense-check whether the answer should be larger or smaller.

Key terms

Similar
Same shape, different size, with equal angles.
Scale factor
The ratio between corresponding lengths.
Corresponding
Matching between the two shapes.
Congruent
Identical in both shape and size.

Related topics

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