Enlargement and similar shapes

Mathematics 2210 · CCEA GCSE · 7 min read · Updated: Aug 31, 2026

Welcome to Enlargement and Similar Shapes!

Have you ever pinched your smartphone screen to zoom in on a photo, or built a scale model of a car? If so, you already understand the basics of enlargement and similar shapes! In this chapter, we will explore how shapes change size while keeping their exact proportions.

Don't worry if geometry sometimes feels confusing. We will break everything down into bite-sized steps so you can tackle exam questions with total confidence.

1. What is an Enlargement?

In everyday English, "enlarge" means to make something bigger. However, in mathematics, an enlargement is a transformation that changes the size of a shape by multiplying all its side lengths by a specific number called the scale factor.

Did you know? In maths, an enlargement can actually make a shape smaller! If the scale factor is a fraction between $0$ and $1$ (like $\frac{1}{2}$), the shape shrinks, but mathematicians still call it an enlargement.

Key Properties of Enlarged Shapes:

• Corresponding angles remain exactly the same: The shape does not bend or distort.
• All side lengths are multiplied by the same scale factor: Every single side grows or shrinks proportionally.
• The new shape and the original shape are mathematically similar.

Quick Summary: Enlargement changes the size of a shape, but preserves its proportions and angles.

2. The Scale Factor ($k$)

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

$$\text{Scale Factor } (k) = \frac{\text{Length on New Shape}}{\text{Corresponding Length on Original Shape}}$$

You can write this simply as: $\text{Scale Factor } = \frac{\text{New Length}}{\text{Old Length}}$

Understanding Different Types of Scale Factors:

• Scale Factor $> 1$ (e.g., $k = 3$): The shape gets larger. Every side is $3$ times longer.
• Fractional Scale Factor between $0$ and $1$ (e.g., $k = \frac{1}{2}$): The shape gets smaller. Every side is halved.
• Negative Scale Factor (e.g., $k = -2$): The shape is enlarged by a factor of $2$, but it is turned upside down and inverted through the centre of enlargement.

Memory Trick: Think of "NO": New over Old to find the scale factor! $\frac{\text{New}}{\text{Old}}$

3. Using a Centre of Enlargement

An enlargement needs two pieces of information:
1. A Scale Factor
2. A Centre of Enlargement (given as coordinates, like $(1, 2)$ or the origin $(0, 0)$)

The centre of enlargement acts like a projector or flashlight. It dictates where the new shape lands on the grid.

Step-by-Step: How to Enlarge a Shape on a Grid

Step 1: Pick one vertex (corner) on the original shape.
Step 2: Count the horizontal and vertical distance from the centre of enlargement to that vertex.
Step 3: Multiply both distances by the scale factor.
Step 4: Starting from the centre of enlargement again, count out your new distances to plot the new vertex.
Step 5: Repeat for all corners and join the dots to complete your shape!

Example: Positive Scale Factor

Enlarge triangle $A$ by scale factor $2$, centre of enlargement $(0, 1)$.
• One corner of triangle $A$ is at $(2, 3)$.
• Distance from centre $(0, 1)$ to $(2, 3)$: $2$ units right, $2$ units up.
• Multiply by scale factor $2$: $2 \times 2 = 4$ units right, $2 \times 2 = 4$ units up.
• Starting from $(0, 1)$, move $4$ right and $4$ up to plot the new point at $(4, 5)$.

Example: Negative Scale Factor

If the scale factor was $-2$:
• Multiply the distance vector by $-2$: $2 \times (-2) = -4$ (meaning $4$ units left), and $2 \times (-2) = -4$ (meaning $4$ units down).
• Starting from the centre $(0, 1)$, go $4$ left and $4$ down to land at $(-4, -3)$.

How to Find the Centre of Enlargement:

If you are given two shapes and asked to find the centre:
1. Draw straight ray lines connecting at least two pairs of matching (corresponding) corners.
2. Extend these lines until they cross.
3. The intersection point is your centre of enlargement!

Key Takeaway: Always count your distances starting from the centre of enlargement, never from the corners of the shape itself.

4. Similar Shapes and Missing Lengths

Two shapes are mathematically similar if one is an enlargement of the other. Their corresponding angles are equal, and their corresponding side lengths are all in the same ratio.

Finding a Missing Side: Step-by-Step

Step 1: Identify a pair of corresponding sides where both lengths are known.
Step 2: Calculate the linear scale factor: $k = \frac{\text{Length of large shape}}{\text{Length of small shape}}$.
Step 3:
• To find a missing side on the larger shape: multiply by $k$.
• To find a missing side on the smaller shape: divide by $k$.

Worked Example:

Triangle $PQR$ is similar to triangle $XYZ$.
• Side $PQ = 6\text{ cm}$ corresponds to side $XY = 18\text{ cm}$.
• Side $QR = 5\text{ cm}$. What is the length of corresponding side $YZ$?

Solution:
1. Scale factor $k = \frac{18}{6} = 3$.
2. To find the larger side $YZ$, multiply: $YZ = 5\text{ cm} \times 3 = 15\text{ cm}$.

5. Area and Volume of Similar Shapes

This is a favourite topic in GCSE exams! When you enlarge a 2D or 3D shape, the area and volume do not just scale up by the length factor.

The Golden Rules of Similarity:

• Length Scale Factor (Linear): $k$
• Area Scale Factor: $k^2$
• Volume Scale Factor: $k^3$

Why does this happen?
Imagine a square of side $2\text{ cm}$. Its area is $2 \times 2 = 4\text{ cm}^2$.
If you double all side lengths to $4\text{ cm}$ (length factor $k = 2$), the new area is $4 \times 4 = 16\text{ cm}^2$.
Notice that the area became $4$ times bigger ($2^2 = 4$), not just $2$ times bigger!

Comparison Table:

• Length Ratio: $a : b$
• Area Ratio: $a^2 : b^2$
• Volume Ratio: $a^3 : b^3$

Worked Example 1: Area

Two similar photos have base lengths of $4\text{ cm}$ and $12\text{ cm}$. The smaller photo has an area of $20\text{ cm}^2$. What is the area of the larger photo?

Solution:
1. Length scale factor $k = \frac{12}{4} = 3$.
2. Area scale factor $= k^2 = 3^2 = 9$.
3. Area of larger photo $= 20\text{ cm}^2 \times 9 = 180\text{ cm}^2$.

Worked Example 2: Volume

Two similar cylinders have heights of $5\text{ cm}$ and $10\text{ cm}$. The larger cylinder has a volume of $400\text{ cm}^3$. Find the volume of the smaller cylinder.

Solution:
1. Length scale factor $k = \frac{10}{5} = 2$.
2. Volume scale factor $= k^3 = 2^3 = 8$.
3. Since we want the smaller cylinder, we divide: $\text{Volume} = \frac{400}{8} = 50\text{ cm}^3$.

Worked Example 3: Going Backwards (From Area/Volume to Length)

Two similar jugs have volumes of $27\text{ cm}^3$ and $64\text{ cm}^3$. If the smaller jug has a height of $9\text{ cm}$, find the height of the larger jug.

Solution:
1. Volume ratio $= 27 : 64$.
2. Take the cube root ($\sqrt[3]{\phantom{x}}$) to get the length ratio:
$\sqrt[3]{27} : \sqrt[3]{64} \implies 3 : 4$.
3. Length scale factor $= \frac{4}{3}$.
4. Height of large jug $= 9 \times \frac{4}{3} = 12\text{ cm}$.

Key Takeaway: Always find the length scale factor ($k$) first! Square it ($k^2$) for area problems, and cube it ($k^3$) for volume problems.

6. Common Mistakes to Avoid

Mistake 1: Multiplying angles by the scale factor.
Correction: Angles never change during an enlargement. If an angle is $35^\circ$, it stays $35^\circ$.

Mistake 2: Counting from the shape instead of the centre of enlargement.
Correction: Always count ray distances starting directly from the given centre point.

Mistake 3: Forgetting to square or cube the scale factor.
Correction: If a question mentions area, use $k^2$. If it mentions volume, capacity, or mass/weight (for objects of uniform density), use $k^3$.

Mistake 4: Getting confused by negative scale factors.
Correction: A negative scale factor just means you measure in the opposite direction through the centre of enlargement.

7. Quick Revision Checklist

Before sitting your exam, make sure you can:
• Enlarge a 2D shape using a positive whole number, fractional, or negative scale factor.
• Find the centre of enlargement using ray lines.
• Identify similar shapes by checking corresponding angles and side ratios.
• Calculate missing lengths in similar triangles and polygons.
• Use the relationships: $\text{Area Factor} = k^2$ and $\text{Volume Factor} = k^3$ to solve practical geometric problems.

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What is a mathematical enlargement and how does it affect angles and side lengths?

An enlargement is a transformation that changes the size of a shape by multiplying all side lengths by a constant scale factor, while keeping corresponding angles unchanged.

How do you calculate the linear scale factor ($k$) between two similar shapes?

$\text{Scale Factor } (k) = \frac{\text{New Length}}{\text{Old Length}}$

How do you enlarge a shape on a coordinate grid using a centre of enlargement and a scale factor?

Count the horizontal and vertical distances from the centre of enlargement to each vertex, multiply those distances by the scale factor, and plot each new vertex measuring from the centre.

How do you perform an enlargement with a negative scale factor?

Multiply the distance vectors from the centre to each vertex by the negative scale factor, and count out the distances in the opposite direction through the centre.

How do you find the centre of enlargement given an original shape and its enlarged image?

Draw straight ray lines through matching (corresponding) vertices of both shapes; the point where all the lines intersect is the centre of enlargement.

How do you find a missing length in similar shapes using the linear scale factor?

Find $k = \frac{\text{known large side}}{\text{known small side}}$, then multiply by $k$ to find a side on the larger shape, or divide by $k$ to find a side on the smaller shape.

What is the relationship between the linear scale factor, area scale factor, and volume scale factor of similar shapes?

If the linear scale factor is $k$, the area scale factor is $k^2$ and the volume scale factor is $k^3$.

How do you find a missing length when given the areas or volumes of two similar shapes?

Take the square root of the area ratio ($\sqrt{\text{area ratio}}$) or the cube root of the volume ratio ($\sqrt[3]{\text{volume ratio}}$) to find the linear scale factor first.

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