Enlargement & Scale Factor –

Enlargement is the fourth type of transformation in GCSE Maths and the only one that changes the size of a shape. It is tested at both Foundation and Higher tiers on AQA, Edexcel, and OCR papers. At Foundation you need to enlarge shapes by positive integer and fractional scale factors. At Higher you must also handle negative scale factors and describe enlargements fully. This guide covers every aspect of the topic, walks you through worked examples at both tiers, flags common errors, and provides practice questions. For the other three transformations, see our reflection, rotation, and translation guide.

What Is an Enlargement?

An enlargement changes the size of a shape while keeping the same proportions. The image is similar to the original — all angles stay the same and all sides are multiplied by the same scale factor.

To fully describe an enlargement you must state:

• The scale factor (how many times bigger or smaller the image is).
• The centre of enlargement (the fixed point from which the shape is enlarged).

Types of Scale Factor

• Scale factor > 1 — the image is larger than the original (e.g. scale factor 3 triples every length).
• Scale factor between 0 and 1 — the image is smaller than the original (e.g. scale factor ½ halves every length). This is sometimes called a reduction.
• Negative scale factor (Higher only) — the image is on the opposite side of the centre of enlargement and inverted. A scale factor of −2 means the image is twice as large and flipped through the centre.

Key Facts

Step-by-Step Method

Performing an Enlargement

1. Mark the centre of enlargement on the grid.
2. Draw a line from the centre to each vertex of the original shape.
3. Multiply the distance from the centre to each vertex by the scale factor.
4. Plot each new vertex at the correct distance from the centre along the same line (or the opposite direction for negative scale factors).
5. Join the new vertices to form the image.

Describing an Enlargement

1. Compare corresponding sides to find the scale factor: new length ÷ original length.
2. To find the centre of enlargement, draw lines through corresponding vertices of the original and image. The point where these lines intersect is the centre.

Finding the Centre of Enlargement

1. Pick two pairs of corresponding vertices.
2. Draw straight lines through each pair, extending them until they meet.
3. The intersection point is the centre of enlargement.

Worked Example 1 — Foundation Level

Question: Enlarge triangle ABC with vertices A(1, 1), B(3, 1), and C(1, 3) by scale factor 2 from the centre of enlargement (0, 0).

Working:

Step 1 — Find the distance of each vertex from the centre (0, 0) and multiply by 2:

• A(1, 1): multiply each coordinate by 2 → A'(2, 2)
• B(3, 1): multiply by 2 → B'(6, 2)
• C(1, 3): multiply by 2 → C'(2, 6)

Step 2 — Plot the new vertices and join them.

Check: Original side AB = 2 units. Image side A'B' = 4 units. Scale factor = 4/2 = 2 ✓

Answer: The image has vertices at A'(2, 2), B'(6, 2), and C'(2, 6).

Worked Example 2 — Higher Level

Question: Shape P is enlarged by scale factor −2 from centre (3, 2). Vertex A of shape P is at (5, 3). Find the position of vertex A' on the image.

Working:

Step 1 — Find the vector from the centre to A: (5 − 3, 3 − 2) = (2, 1)

Step 2 — Multiply by the scale factor −2: (2 × −2, 1 × −2) = (−4, −2)

Step 3 — Add this to the centre: A' = (3 + (−4), 2 + (−2)) = (−1, 0)

Answer: A' is at (−1, 0). The image point is on the opposite side of the centre and twice as far away.

Common Mistakes

• Forgetting to state the centre of enlargement. The description "enlargement by scale factor 2" is incomplete and will lose marks. You must give the centre.
• Measuring distances from the shape instead of from the centre. All distances must be measured from the centre of enlargement.
• Confusing scale factor with area factor. If the scale factor is 3, lengths are 3 times bigger but the area is 9 times bigger (3²). Do not mix these up.
• Drawing lines that do not pass through the centre. When checking your work, lines through corresponding vertices should all pass through the centre.
• Negative scale factor direction. For a negative scale factor, the image appears on the opposite side of the centre. Forgetting to go through the centre to the other side is a common Higher tier error.

Exam Tips

• For fractional scale factors, the image is smaller. If the scale factor is ⅓, every length is divided by 3. The image will be closer to the centre.
• Always draw construction lines from the centre through each vertex. This helps you place the image accurately and shows your method.
• To find the scale factor from a diagram, divide an image length by the corresponding original length. If the answer is less than 1, the shape has been reduced.
• Area and volume relationships are popular at Higher level. Remember: area factor = (scale factor)² and volume factor = (scale factor)³.
• Use column vectors to handle distances from the centre — this is the most reliable method and avoids counting errors on the grid.

Practice Questions

Question 1: Enlarge the point (2, 3) by scale factor 3 from the origin.

Question 2: A shape is enlarged by scale factor 2. The original area is 12 cm². What is the area of the image?

Question 3: A triangle has vertices at (1, 1), (5, 1), and (1, 4). It is enlarged by scale factor ½ from the point (1, 1). Find the image vertices.

Question 4: Describe fully the single transformation that maps a triangle with side 4 cm to a similar triangle with corresponding side 12 cm, given the centre is at the origin.

Question 5: A shape is enlarged by scale factor −1 from the point (2, 3). Vertex P is at (5, 7). Find the image of P.

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Related Topics

• Transformations: Reflection, Rotation, Translation — the three congruence transformations.
• Congruence and Similarity — similar shapes and enlargement are closely linked.
• Vectors — using vectors to describe movements from the centre.
• Plans and Elevations — understanding shapes from different viewpoints.

Summary

Enlargement changes the size of a shape by a scale factor from a centre of enlargement. A scale factor greater than 1 makes the shape bigger, a fractional scale factor makes it smaller, and a negative scale factor (Higher only) inverts the shape through the centre. To perform an enlargement, measure distances from the centre, multiply by the scale factor, and plot the new vertices. To describe one, state the type (enlargement), the scale factor, and the centre. Remember that area scales by the square of the scale factor and volume by the cube. These questions appear regularly and carry reliable marks for students who show clear, methodical working.