Similarity

Q: What two mathematical conditions must be satisfied for two shapes to be similar?

Corresponding angles must be equal, and the ratios of corresponding side lengths must be equal (meaning all sides are multiplied by the same scale factor).

Q: How can you prove that two triangles are mathematically similar?

You only need to show that two pairs of corresponding angles are equal (Angle-Angle / AAA rule), as the third pair must also be equal.

Q: How do you identify which sides correspond to each other in two similar triangles?

Corresponding sides are always opposite the equal angles in the respective triangles.

Q: How do you calculate unknown side lengths when two similar triangles are nested inside one another?

Use the full side length of the larger triangle (the sum of the segments) to find the LSF, then multiply the corresponding original length by the LSF.

Q: What is the relationship between the Length Scale Factor (\(k\)) and the Area Scale Factor (ASF)?

\(\text{ASF} = (\text{LSF})^2 = k^2\), so \(\text{Area}_{\text{New}} = \text{Area}_{\text{Original}} \times k^2\).

Q: What is the relationship between the Length Scale Factor (\(k\)) and the Volume Scale Factor (VSF)?

\(\text{VSF} = (\text{LSF})^3 = k^3\), so \(\text{Volume}_{\text{New}} = \text{Volume}_{\text{Original}} \times k^3\).

Q: How do you convert directly between area and volume ratios of similar shapes?

Always find the linear scale factor \(k\) first: \(k = \sqrt{\text{ASF}}\) or \(k = \sqrt[3]{\text{VSF}}\), then calculate the desired scale factor (\(k^3\) for volume or \(k^2\) for area).

Practice

Mathematics (Specification A) · Pearson Edexcel IGCSE · 6 min read · Updated: Sep 13, 2026

📐 Geometry Chapter: Similarity – The Art of Scaling 📐

Hello future mathematicians! Welcome to the chapter on Similarity. Don't worry if geometry sometimes feels like drawing complicated pictures—this topic is incredibly logical and practical.

In this chapter, you'll learn how to compare shapes that look exactly the same but have been perfectly scaled up or down. Think of it like using the zoom feature on your phone camera! Mastering similarity is crucial for understanding maps, blueprints, and 3D geometry problems. Let's dive in!

1. Defining Similarity: Same Shape, Different Size

What does "Similar" mean in Mathematics?

In everyday language, "similar" means almost the same. But in maths, similarity has a precise meaning:

Two shapes are similar if they have exactly the same shape but potentially a different size.
One shape is an enlargement or reduction of the other.

Crucial Requirement: For two shapes to be similar, two things MUST be true:

Corresponding Angles must be Equal. If Shape A has an angle of \(50^\circ\), the matching angle (the corresponding angle) in Similar Shape B must also be \(50^\circ\).
Ratios of Corresponding Side Lengths must be Equal. This means all pairs of corresponding sides must be multiplied (or divided) by the exact same number. This number is called the Scale Factor.

Did you know? Similarity is different from Congruence. Congruent shapes are identical in both shape AND size (a perfect copy). Similar shapes only need to share the shape.

Quick Review: Similarity requires matching angles and a consistent scale factor for all sides.

2. Focusing on Similar Triangles

Triangles are the most common shapes used to test similarity in exams. Luckily, triangles have a very simple rule for proving similarity!

How to Prove Two Triangles are Similar

If you can prove that all three pairs of corresponding angles are equal, the triangles are automatically similar.

The Angle Rule (AAA):

If Angle 1 in Triangle A = Angle 1 in Triangle B,
And Angle 2 in Triangle A = Angle 2 in Triangle B,
Then, the third angles must also be equal (since the angles in a triangle always sum to \(180^\circ\)).

Therefore, to prove similarity in triangles, you only need to show that two pairs of corresponding angles are equal. This is often the quickest method!

Identifying Corresponding Sides

When triangles are placed strangely or overlap, it can be tricky to figure out which sides match up (correspond).

Memory Aid: A side corresponds to the side opposite the equal angle in the other triangle.

Example: If side x is opposite the \(60^\circ\) angle in the small triangle, then the corresponding side y must be opposite the \(60^\circ\) angle in the large triangle.
Tip: If the shapes overlap, always try to draw them separately and reorient them so they face the same direction. This makes matching sides much easier!

Key Takeaway: Focus on matching angles first. Once angles are matched, the sides opposite those angles are the corresponding sides you need for calculations.

3. Calculating and Using the Length Scale Factor (LSF)

The Length Scale Factor (LSF) is the constant multiplier that links all corresponding side lengths of two similar shapes.

Finding the Length Scale Factor (LSF)

The LSF (often written as \(k\)) is calculated by finding the ratio of two known corresponding sides:

\(k = \text{LSF} = \frac{\text{Length of side in New Shape}}{\text{Length of corresponding side in Original Shape}}\)

Example Calculation:

Suppose a side in the small shape is \(4\text{ cm}\), and the corresponding side in the large shape is \(12\text{ cm}\).
\(\text{LSF} = \frac{12}{4} = 3\)
This means lengths in the large shape are \(3\) times those in the small shape.

Important Consistency Check:
You must decide which shape is 'New' (or 'Big') and stick to that choice throughout the problem.

If \(\text{LSF} > 1\), it is an Enlargement.
If \(\text{LSF} < 1\), it is a Reduction.

Step-by-Step: Finding Unknown Side Lengths

Let's say you have two similar triangles and need to find the unknown length \(x\):

Identify Corresponding Sides: Look for the pair of sides that are both known (e.g., side \(8\text{ cm}\) corresponds to side \(10\text{ cm}\)).
Calculate the LSF: If scaling from small to large: \(\text{LSF} = \frac{10}{8} = 1.25\)
Apply the LSF: Multiply the corresponding known side by the LSF.
If the side corresponding to \(x\) is \(6\text{ cm}\) in the small shape: \(x = 6 \times 1.25 = 7.5\text{ cm}\)

Common Mistake Alert!

When two similar shapes are nested (one inside the other, sharing a vertex), students often confuse the side lengths.

The side length of the BIG triangle is the WHOLE length, not just the added segment!

Example: If the small triangle side is \(5\text{ cm}\), and the added segment below it is \(3\text{ cm}\), the whole side of the large triangle is \(5 + 3 = 8\text{ cm}\). Use \(5\) and \(8\) to calculate the LSF.

4. Scaling Up to Area and Volume (Higher Tier)

When working with 2D areas or 3D volumes of similar figures, you cannot use the Length Scale Factor directly. Length, area, and volume scale in different dimensions:

Length Scale Factor (1D): \(k\)
Area Scale Factor (2D): \(k^2\)
Volume Scale Factor (3D): \(k^3\)

The Area Scale Factor (ASF)

If the ratio of lengths is \(k\), the ratio of areas is \(k^2\):
\(\text{ASF} = (\text{LSF})^2 = k^2\)
\(\text{Area}_{\text{New}} = \text{Area}_{\text{Original}} \times k^2\)

Why? Imagine a \(2\text{ cm} \times 2\text{ cm}\) square with area \(4\text{ cm}^2\). If you double the lengths (\(k = 2\)), the new square is \(4\text{ cm} \times 4\text{ cm}\), giving an area of \(16\text{ cm}^2\). The area is \(4\) times bigger, which is \(2^2\).

Example (Area): Two similar shapes have corresponding side lengths of \(3\text{ cm}\) and \(6\text{ cm}\). If the smaller shape has an area of \(15\text{ cm}^2\):

\(\text{LSF} = k = \frac{6}{3} = 2\)
\(\text{ASF} = k^2 = 2^2 = 4\)
\(\text{Area}_{\text{Large}} = 15 \times 4 = 60\text{ cm}^2\)

The Volume Scale Factor (VSF)

If the ratio of lengths is \(k\), the ratio of volumes of similar 3D solids is \(k^3\):
\(\text{VSF} = (\text{LSF})^3 = k^3\)
\(\text{Volume}_{\text{New}} = \text{Volume}_{\text{Original}} \times k^3\)

Why? Imagine a \(2\text{ cm} \times 2\text{ cm} \times 2\text{ cm}\) cube with volume \(8\text{ cm}^3\). Doubling the side lengths (\(k = 2\)) gives a \(4\text{ cm} \times 4\text{ cm} \times 4\text{ cm}\) cube with volume \(64\text{ cm}^3\). The volume is \(8\) times bigger, which is \(2^3\).

Example (Volume): Two similar cylinders have heights of \(5\text{ cm}\) and \(15\text{ cm}\). The volume of the smaller cylinder is \(40\text{ cm}^3\). Find the volume of the larger cylinder:

\(\text{LSF} = k = \frac{15}{5} = 3\)
\(\text{VSF} = k^3 = 3^3 = 27\)
\(\text{Volume}_{\text{Large}} = 40 \times 27 = 1080\text{ cm}^3\)

⚠️ Moving Between Length, Area, and Volume ⚠️

Always find the Length Scale Factor \(k\) first before converting between area and volume:

Given Areas: Find \(k = \sqrt{\text{ASF}}\), then cube it to find \(\text{VSF} = k^3\).
Given Volumes: Find \(k = \sqrt[3]{\text{VSF}}\), then square it to find \(\text{ASF} = k^2\).

5. Review and Encouragement

Quick Chapter Summary

Relationship	Scale Factor	Formula
Lengths / Perimeters	\(k\)	\(\text{New Length} = \text{Original Length} \times k\)
Surface Areas / Areas	\(k^2\)	\(\text{New Area} = \text{Original Area} \times k^2\)
Volumes / Capacities	\(k^3\)	\(\text{New Volume} = \text{Original Volume} \times k^3\)

Don't worry if this seems tricky at first! Similarity relies on setting up the correct scale factors. Identify the matching sides, determine \(k\), and apply \(k\), \(k^2\), or \(k^3\) depending on whether the problem asks for length, area, or volume. Keep practicing and you will master this topic in no time!

References

Pearson Edexcel International GCSE in Mathematics (Specification A) Specification — Pearson Edexcel

Quick check

Can you answer these now?

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

What two mathematical conditions must be satisfied for two shapes to be similar?

Corresponding angles must be equal, and the ratios of corresponding side lengths must be equal (meaning all sides are multiplied by the same scale factor).

How can you prove that two triangles are mathematically similar?

You only need to show that two pairs of corresponding angles are equal (Angle-Angle / AAA rule), as the third pair must also be equal.

How do you identify which sides correspond to each other in two similar triangles?

Corresponding sides are always opposite the equal angles in the respective triangles.

How do you calculate the Length Scale Factor (LSF or \(k\)) between two similar shapes?

\(k = \text{LSF} = \frac{\text{Length of side in New Shape}}{\text{Length of corresponding side in Original Shape}}\)

How do you calculate unknown side lengths when two similar triangles are nested inside one another?

Use the full side length of the larger triangle (the sum of the segments) to find the LSF, then multiply the corresponding original length by the LSF.

What is the relationship between the Length Scale Factor (\(k\)) and the Area Scale Factor (ASF)?

\(\text{ASF} = (\text{LSF})^2 = k^2\), so \(\text{Area}_{\text{New}} = \text{Area}_{\text{Original}} \times k^2\).

What is the relationship between the Length Scale Factor (\(k\)) and the Volume Scale Factor (VSF)?

\(\text{VSF} = (\text{LSF})^3 = k^3\), so \(\text{Volume}_{\text{New}} = \text{Volume}_{\text{Original}} \times k^3\).

How do you convert directly between area and volume ratios of similar shapes?

Always find the linear scale factor \(k\) first: \(k = \sqrt{\text{ASF}}\) or \(k = \sqrt[3]{\text{VSF}}\), then calculate the desired scale factor (\(k^3\) for volume or \(k^2\) for area).

Ready to test yourself?

9 practice questions on this topic, with worked solutions. Multiple choice is marked as you go. No account needed.

Practise This Topic

More Mathematics (Specification A) chapters

* The content provided by thinka is generated by AI and may not always be accurate or up-to-date. Please use it as a supplementary resource and verify with official materials.

Put These Notes into Practice

Reading the notes is a great start. Now practice with unlimited AI-generated questions and get instant feedback. 100,000+ students are already improving their grades.

Start Practicing Now View Pricing

Done reading? Test yourself with AI practice questions

Practice This Topic Now