Enlargement and Scale Factor: KS3 Maths Guide

An enlargement changes the size of a shape by a scale factor without altering its angles. Every length in the enlarged shape equals the original length multiplied by the scale factor. This is core KS3 geometry content tested in Year 7, 8, and 9.

What is a scale factor?

The scale factor tells you how many times bigger (or smaller) the enlarged image is compared to the original shape.

Scale factor > 1 → image is larger than the original.

Scale factor = 1 → image is identical in size.

Scale factor between 0 and 1 (a fraction) → image is smaller than the original.

Scale factor < 0 (negative) → image is on the opposite side of the centre of enlargement and flipped through 180° (GCSE extension, sometimes introduced at KS3).

Formula:

Scale factor = image length ÷ original length

Or rearranged to find a missing image length:

Image length = original length × scale factor

How to find a scale factor from two similar shapes

Worked example 1: calculate the scale factor

A rectangle has width 4 cm and length 6 cm. An enlarged rectangle has width 10 cm and length 15 cm. Find the scale factor.

Using the widths:

Scale factor = 10 ÷ 4 = 2.5

Check using the lengths:

Scale factor = 15 ÷ 6 = 2.5 ✓

Answer: scale factor = 2.5

Worked example 2: find a missing length

Triangle A has sides 3 cm, 4 cm, and 5 cm. It is enlarged with scale factor 3 to give triangle B. Find the side lengths of triangle B.

3 × 3 = 9 cm
4 × 3 = 12 cm
5 × 3 = 15 cm

Answer: triangle B has sides 9 cm, 12 cm, and 15 cm.

Check: 9 ÷ 3 = 3, 12 ÷ 3 = 4, 15 ÷ 3 = 5 — matches the original ✓

Worked example 3: scale factor less than 1 (reduction)

A photograph is 20 cm wide. A thumbnail version is 5 cm wide. What is the scale factor?

Scale factor = 5 ÷ 20 = 0.25

This is a scale factor of 0.25 (or ¼). The thumbnail is one quarter of the original width.

How to enlarge a shape from a centre of enlargement

When a centre of enlargement is given, each point of the image is placed at a specific position relative to the centre, not just scaled in size.

Steps to enlarge from a centre of enlargement

Draw a ray from the centre of enlargement through each vertex (corner) of the original shape.

Measure the distance from the centre to each vertex.

Multiply each distance by the scale factor to get the distance from the centre to the corresponding image vertex.

Mark the image vertices and join them up.

Worked example 4: enlargement on a grid

Shape ABCD has vertices at A(1, 1), B(3, 1), C(3, 2), D(1, 2). The centre of enlargement is at O(0, 0) and the scale factor is 2.

Multiply every coordinate by the scale factor (works when the centre is at the origin):

Original vertex | Calculation | Image vertex
A(1, 1) | × 2 | A′(2, 2)
B(3, 1) | × 2 | B′(6, 2)
C(3, 2) | × 2 | C′(6, 4)
D(1, 2) | × 2 | D′(2, 4)

Answer: image vertices are A′(2, 2), B′(6, 2), C′(6, 4), D′(2, 4).

Check: original width AB = 3 − 1 = 2. Image width A′B′ = 6 − 2 = 4. 4 ÷ 2 = 2 ✓ (matches scale factor)

Worked example 5: centre of enlargement not at the origin

Triangle PQR has P(3, 2), Q(5, 2), R(5, 4). The centre of enlargement is C(1, 0) and the scale factor is 2.

Step 1 — Find the vector from the centre to each vertex:

P: (3 − 1, 2 − 0) = (2, 2)

Q: (5 − 1, 2 − 0) = (4, 2)

R: (5 − 1, 4 − 0) = (4, 4)

Step 2 — Multiply each vector by the scale factor:

P′ vector: (2 × 2, 2 × 2) = (4, 4) → P′ = (1 + 4, 0 + 4) = (5, 4)

Q′ vector: (4 × 2, 2 × 2) = (8, 4) → Q′ = (1 + 8, 0 + 4) = (9, 4)

R′ vector: (4 × 2, 4 × 2) = (8, 8) → R′ = (1 + 8, 0 + 8) = (9, 8)

Answer: P′(5, 4), Q′(9, 4), R′(9, 8).

Negative scale factor (extension)

A negative scale factor means the image is on the opposite side of the centre of enlargement, rotated 180°. The size is still scaled by the magnitude (absolute value) of the factor.

Example: Scale factor −2 from centre O(0, 0) on point A(3, 1):

A′ = (3 × −2, 1 × −2) = (−6, −2)

The image is twice as large and on the other side of the origin.

How areas change under enlargement

When a shape is enlarged by scale factor k:

Lengths are multiplied by k.

Areas are multiplied by k².

Worked example 6: area after enlargement

A square has side length 5 cm (area = 25 cm²). It is enlarged with scale factor 3.

New side length: 5 × 3 = 15 cm
New area: 15 × 15 = 225 cm²

Check using the area multiplier: 25 × 3² = 25 × 9 = 225 cm² ✓

This is a key GCSE link introduced at the top of KS3.

How enlargement fits the KS3 national curriculum

The Department for Education's KS3 Mathematics Programme of Study requires pupils to "identify properties of and describe the results of translations, rotations and reflections applied to given figures" and to understand scale. Enlargement, including use of a centre of enlargement, is listed explicitly as a KS3 transformation alongside rotation, reflection, and translation. Scale factor, similar shapes, and enlargement from a given centre are core Year 8 and Year 9 content within the statutory national curriculum for Key Stage 3 and 4.

Common mistakes to avoid

Mistake 1 — Multiplying only one side of a shape.
Every length must be multiplied by the scale factor — not just the base or just the height.

Mistake 2 — Forgetting the centre of enlargement.
If a centre is given, the position of the image on the grid depends on measuring from the centre. Ignoring the centre gives the right shape but the wrong position.

Mistake 3 — Confusing scale factor for areas and lengths.
If the scale factor is 3, areas are multiplied by 9 (not 3). Lengths scale by k; areas scale by k².

Mistake 4 — Reversing the formula.
Scale factor = image ÷ original, not original ÷ image. Double-check which shape is the original.

Frequently asked questions

What does a scale factor of 0.5 mean?

A scale factor of 0.5 means the image is half the size of the original — every length is halved. Despite being called an "enlargement", a scale factor between 0 and 1 actually reduces the shape. The mathematical term still applies.

How do you find the centre of enlargement from two similar shapes?

Draw straight lines (rays) through pairs of corresponding vertices — one vertex from the original and the matching vertex from the image. The point where all these rays meet is the centre of enlargement. If the shapes are given on a coordinate grid, you can extend the lines until they intersect at a single point.

Are corresponding angles changed by enlargement?

No. Enlargement changes lengths but preserves all angles. This is why an enlarged shape is described as similar to the original — same shape, different size. If angles were also changed, the shape would be distorted rather than scaled.

How is scale factor used outside maths lessons?

Architects draw buildings to scale (for example, 1 cm on a plan = 50 cm in real life, a scale factor of 50 the other way). Map makers reduce real distances by a scale factor printed as a ratio (1:25 000 on an Ordnance Survey map). Photographers enlarge or reduce images on screen using the same principle.

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