Teaching unit · UK Year 9 (Key Stage 3, ages 13 to 14)

Pythagoras' theorem

Finding the hypotenuse, finding a missing leg, and solving real-world right-triangle problems

About three lessons of 45 to 60 minutes

Start here · hook

The 4,000-year-old rule that still builds every staircase

Ancient Egyptian rope-stretchers tied 12 evenly spaced knots into a loop of rope and pulled it into a triangle with sides 3, 4 and 5 knots long, guaranteeing a perfect right angle every time, no protractor needed. That trick, and the deeper rule behind it, is thousands of years old, and it is still exactly how builders check a wall is truly square today.

The rule is Pythagoras' theorem: in any right triangle, a squared plus b squared equals c squared, where c is always the hypotenuse, the longest side, directly opposite the right angle. It works for every right triangle, any size, and it lets you find a missing side you could never directly measure, whether that is a ladder's safe reach, a TV's true diagonal size, or the shortest path across a park.

A rope loop with sides 3, 4 and 5 knotsthe ancient Egyptian trick for a guaranteed right angle: $3^{2}$ + $4^{2}$ = 9 + 16 = 25 = $5^{2}$
A ladder leaning against a wallthe ladder is the hypotenuse; the wall height and ground distance are the 2 legs
A TV's advertised screen sizethe diagonal, found from the width and height using Pythagoras' theorem
A cyclist's straight-line distance from homefound from how far east and how far north she rode, even though she never rode in a straight line

Learning objective

What students will be able to do

Students will identify the hypotenuse of a right triangle, use $a^{2}$ + $b^{2}$ = $c^{2}$ to find the hypotenuse when both legs are known, rearrange the theorem to find a missing leg when the hypotenuse and 1 leg are known, and apply Pythagoras' theorem to real-world right-triangle problems.

Success criteria

I can identify the hypotenuse of a right triangle.
I can use $a^{2}$ + $b^{2}$ = $c^{2}$ to find the hypotenuse when both legs are known.
I can rearrange the theorem to find a missing leg when the hypotenuse and 1 leg are known.
I can recognise a Pythagorean triple, such as 3-4-5 or 5-12-13.
I can apply Pythagoras' theorem to a real-world right-triangle problem.
I can explain why the final step of every calculation is a square root.

Curriculum anchor

Standards this unit teaches

KS3 Maths: Geometry and measuresUK National Curriculum (England)
Geometry and measures
Statutory requirement (Department for Education, "National curriculum in England: mathematics programmes of study", updated 28 September 2021, Key stage 3, "Geometry and measures" strand, https://www.gov.uk/government/publications/national-curriculum-in-england-mathematics-programmes-of-study/national-curriculum-in-england-mathematics-programmes-of-study): pupils should be taught to "apply angle facts, triangle congruence, similarity and properties of quadrilaterals to derive results about angles and sides, including Pythagoras’ Theorem, and use known results to obtain simple proofs"; "use Pythagoras’ Theorem and trigonometric ratios in similar triangles to solve problems involving right-angled triangles". This unit builds the Pythagoras’ Theorem half of that second bullet; trigonometric ratios are a distinct, larger topic for a future unit.

Before you start

Prior knowledge

This unit builds on skills students should already have met. Revisit any that are shaky first.

Key vocabulary

Words to teach and display

Hypotenuse: the longest side of a right triangle, always directly opposite the right angle
Leg: either of the 2 shorter sides of a right triangle, the 2 sides that form the right angle
Pythagorean triple: a set of 3 whole numbers that satisfy $a^{2}$ + $b^{2}$ = $c^{2}$ exactly, such as 3-4-5 or 5-12-13
Right angle: an angle of exactly 90 degrees, usually marked with a small square in a diagram

Teaching sequence

Teach it: concrete, pictorial, abstract

The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.

1. Finding the hypotenuse

Concrete

In any right triangle, the square of the hypotenuse equals the sum of the squares of the other 2 sides: $a^{2}$ + $b^{2}$ = $c^{2}$ , where c is always the hypotenuse, the side directly opposite the right angle.

A right triangle with legs 6 and 8 always has a hypotenuse of exactly 10, because $6^{2}$ + $8^{2}$ = 36 + 64 = 100, and the square root of 100 is 10. This works for every right triangle, any size, which is why sets like 3-4-5, 6-8-10 and 5-12-13 (Pythagorean triples) come up again and again.

A right triangle with legs 6 and 8. The hypotenuse, computed with

a^{2}

b^{2}

c^{2}

, is exactly 10.

Worked example

A right triangle has legs of length 9 and 12. Find the hypotenuse.

Apply the theorem: $c^{2}$ = $a^{2}$ + $b^{2}$ = $9^{2}$ + $12^{2}$ .
$9^{2}$ + $12^{2}$ = 81 + 144 = 225.
c = $\sqrt{225}$ = 15.

Answer: The hypotenuse is 15.

Check for understanding, ask

Which side is always labelled c, and how do you spot it in a diagram?
Why is the last step a square root rather than a division?

2. Finding a missing leg

Pictorial

If the hypotenuse and 1 leg are already known, rearrange the theorem: a missing leg, squared, equals the hypotenuse squared minus the known leg squared. This time SUBTRACT, since the hypotenuse is always the biggest side already.

A classic real-world case: a ladder 13 m long leans against a wall with its base 5 m from the wall. The ladder is the hypotenuse (the longest side, opposite the right angle at the ground). The height up the wall is the missing leg: $\sqrt{13^{2} - 5^{2}}$ = $\sqrt{169 - 25}$ = $\sqrt{144}$ = 12.

A ladder 13 m long with its base 5 m from the wall reaches 12 m up the wall (5-12-13 is a Pythagorean triple).

Worked example

A right triangle has a hypotenuse of 17 and 1 leg of 8. Find the other leg.

leg² = hypotenuse² - known leg² = $17^{2}$ - $8^{2}$ = 289 - 64 = 225.
leg = $\sqrt{225}$ = 15.

Answer: The missing leg is 15.

Check for understanding, ask

Why does finding a missing leg SUBTRACT the squares instead of adding them?
In the ladder problem, which measurement is the hypotenuse, and how do you know?

3. Real-world right-triangle problems

Abstract

Many real situations hide a right triangle, even when no triangle is drawn: a diagonal brace across a rectangular gate, a TV screen's advertised size, or the straight-line distance between 2 points on a map described by an east-west distance and a north-south distance.

A cyclist rides 9 km east, then 12 km north. She never travelled in a straight line, but her straight-line distance from her starting point is the hypotenuse of a right triangle with legs 9 and 12: $\sqrt{9^{2} + 12^{2}}$ = $\sqrt{81 + 144}$ = $\sqrt{225}$ = 15 km.

A TV screen 32 inches wide and 24 inches tall has an advertised diagonal size of exactly 40 inches (12-16-20 scaled up).

Worked example

A cyclist rides 9 km east, then 12 km north. Find her straight-line distance from her starting point.

The east and north distances are the 2 legs of a right triangle; the straight-line distance is the hypotenuse.
distance² = $9^{2}$ + $12^{2}$ = 81 + 144 = 225.
distance = $\sqrt{225}$ = 15 km.

Answer: She is 15 km from her starting point.

Check for understanding, ask

List 2 other real-world situations where Pythagoras' theorem could find a distance you cannot measure directly.
In the cyclist problem, which distance represents the hypotenuse, and why?

Watch for

Common misconceptions and how to address them

MisconceptionWhen finding a missing leg, add the squares instead of subtracting, using $c^{2}$ = $a^{2}$ + $b^{2}$ even when c is not the unknown side.

Why it happens: Students memorise 'square, square, square root' from the hypotenuse case without first checking which side is actually missing.

How to address it: Always identify which side is unknown FIRST. If the HYPOTENUSE is missing, ADD the squares of the 2 legs. If a LEG is missing, SUBTRACT the known leg's square from the hypotenuse's square (the hypotenuse is always the largest side, so what is left after subtracting is always positive).

MisconceptionComputing $c^{2}$ correctly, but writing that value down as the final answer, e.g. leaving 100 as the hypotenuse instead of 10.

Why it happens: The theorem gives you c SQUARED, not c itself, and it is easy to stop 1 step too early.

How to address it: $a^{2}$ + $b^{2}$ = $c^{2}$ gives the square of the length you want. Always finish with a square root to get the actual side length: $\sqrt{100}$ = 10, not 100.

MisconceptionAssuming any 3 side lengths that are 'roughly the right size' form a right triangle, without checking the numbers exactly.

Why it happens: Without testing $a^{2}$ + $b^{2}$ = $c^{2}$ numerically, students judge by a rough visual impression rather than an exact calculation.

How to address it: Test it exactly: for sides 5, 6 and 8, $5^{2}$ + $6^{2}$ = 25 + 36 = 61, but $8^{2}$ = 64. Since 61 does not equal 64, 5-6-8 is NOT a right triangle. Only sides that satisfy $a^{2}$ + $b^{2}$ = $c^{2}$ exactly, like 3-4-5, work.

MisconceptionApplying Pythagoras' theorem to a triangle that is not right-angled.

Why it happens: Pythagoras' theorem only holds for right triangles; without checking for the right angle, students sometimes apply the formula to any triangle with 3 roughly-known sides.

How to address it: Always confirm the right angle first (it is usually marked with a small square in a diagram, or stated directly). Without a right angle, $a^{2}$ + $b^{2}$ = $c^{2}$ simply does not hold, and a different method is needed.

Do it together

Guided practice (with answers)

1. A right triangle has legs 5 and 12. Find the hypotenuse.
Answer: 13, because $5^{2}$ + $12^{2}$ = 25 + 144 = 169, and $\sqrt{169}$ = 13.
2. A right triangle has legs 7 and 24. Find the hypotenuse.
Answer: 25, because $7^{2}$ + $24^{2}$ = 49 + 576 = 625, and $\sqrt{625}$ = 25.
3. A right triangle has a hypotenuse of 41 and 1 leg of 9. Find the other leg.
Answer: 40, because $41^{2}$ - $9^{2}$ = 1681 - 81 = 1600, and $\sqrt{1600}$ = 40.
4. A right triangle has a hypotenuse of 29 and 1 leg of 20. Find the other leg.
Answer: 21, because $29^{2}$ - $20^{2}$ = 841 - 400 = 441, and $\sqrt{441}$ = 21.
5. A ladder 10 m long leans against a wall with its base 6 m from the wall. How high does it reach?
Answer: 8 m, because $10^{2}$ - $6^{2}$ = 100 - 36 = 64, and $\sqrt{64}$ = 8.
6. Is a triangle with sides 6, 8 and 11 a right triangle? Explain.
Answer: No, because $6^{2}$ + $8^{2}$ = 36 + 64 = 100, but $11^{2}$ = 121. Since 100 does not equal 121, it is not a right triangle.

On their own

Independent practice worksheets

Practise finding the hypotenuse, finding a missing leg, and solving real-world problems with computed, never-wrong answer keys, using whole-number Pythagorean triples so every triangle answer is exact.

Year 9 Pythagoras' theorem: finding the hypotenuse worksheetthe core skill from section 1 Year 9 Pythagoras' theorem: finding a missing leg worksheetthe core skill from section 2 Year 9 Pythagoras' theorem: word problems worksheetreal-world applications, the core skill from section 3

Reach every student

Differentiation

Support

Always draw and label the triangle first, marking the right angle, before writing any equation.
Keep a written checklist visible: 1) identify the right angle, 2) identify what is missing (hypotenuse or a leg), 3) add or subtract the squares accordingly, 4) square root at the end.
Start with only 'find the hypotenuse' problems using the 3-4-5 triple before introducing missing-leg problems.
Colour the hypotenuse a different colour from the 2 legs in every diagram, so it is never confused with a leg.

Extension

Investigate other Pythagorean triples beyond 3-4-5 and 5-12-13, such as 8-15-17, 7-24-25 and 20-21-29, and check each satisfies $a^{2}$ + $b^{2}$ = $c^{2}$ exactly.
Solve a 2-step problem that needs Pythagoras' theorem twice, such as finding the diagonal of a cuboid from its length, width and height.
Explore why scaling every side of a right triangle by the same factor (e.g. doubling) also scales the hypotenuse by that factor.
Introduce the converse: given 3 side lengths, decide whether the triangle is right-angled, acute or obtuse by comparing $c^{2}$ to $a^{2}$ + $b^{2}$ .

Check it stuck

Assessment: exit ticket

A three-question exit ticket sampling finding the hypotenuse, finding a missing leg, and a real-world application.

1. A right triangle has legs 8 and 15. Find the hypotenuse.
Answer: 17, because $8^{2}$ + $15^{2}$ = 64 + 225 = 289, and $\sqrt{289}$ = 17.
2. A right triangle has a hypotenuse of 25 and 1 leg of 7. Find the other leg.
Answer: 24, because $25^{2}$ - $7^{2}$ = 625 - 49 = 576, and $\sqrt{576}$ = 24.
3. A rectangular sports pitch is 40 m long and 30 m wide. Find the length of its diagonal.
Answer: 50 m, because $40^{2}$ + $30^{2}$ = 1600 + 900 = 2500, and $\sqrt{2500}$ = 50.

For the teacher

Teacher notes and timings

Rough timing across 3 lessons: Lesson 1 finding the hypotenuse (section 1), Lesson 2 finding a missing leg (section 2), Lesson 3 real-world problems plus the exit ticket (section 3).
This unit deliberately covers only the Pythagoras' Theorem half of the curriculum's combined bullet ('Pythagoras' Theorem and trigonometric ratios in similar triangles'); trigonometric ratios (sine, cosine, tangent) are a distinct, larger topic reserved for a future unit, the same scoping choice the existing grade-8-pythagorean-theorem-volume unit makes.
Every worksheet item uses a genuine whole-number Pythagorean triple (optionally scaled), so every hypotenuse and every missing leg is an EXACT integer, never a rounded decimal, keeping the 'never wrong' answer-key guarantee intact even for square-root-based answers.
Language to keep repeating: identify the right angle and the hypotenuse first; add the squares to find the hypotenuse, subtract to find a leg; always finish with a square root.
Use Student view to project this lesson. Print saves the full teacher unit, including answers and teacher notes; use the linked independent-practice worksheets for student handouts.

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