Pythagoras’ Theorem: The Complete O-Level Guide

One of the most famous theorems in mathematics — and a must-know for O-Level Math. Learn to find missing sides in any right-angled triangle.

$a^2 + b^2 = c^2$

What is Pythagoras’ Theorem?

Pythagoras’ Theorem states that in any right-angled triangle, the square of the hypotenuse (the longest side) equals the sum of the squares of the other two sides.

The Formula:

a^2 + b^2 = c^2

Where c is the hypotenuse and a, b are the two shorter sides (legs).

This theorem only works for right-angled triangles — triangles that have exactly one 90° angle.

Step 1: Identify the Hypotenuse

Before using the theorem, you must identify which side is the hypotenuse:

The Hypotenuse is…

The longest side
Always opposite the right angle
Never touches the 90° corner

The Two Legs are…

The two shorter sides
They form the right angle
Both touch the 90° corner

Example: Labeling a Right Triangle

In triangle ABC below, the right angle is at C. The hypotenuse is the side opposite C — that’s side AB.

💡 Key Point

The hypotenuse is always labeled c in the formula. The two legs can be labeled a and b in either order.

Pythagorean Triples: Numbers Worth Memorizing

Pythagorean triples are sets of three whole numbers that satisfy $a^2 + b^2 = c^2$ . Memorizing common triples helps you solve problems faster.

Triple (a, b, c)	Verification	Multiples
3, 4, 5	9 + 16 = 25	6-8-10, 9-12-15, 12-16-20
5, 12, 13	25 + 144 = 169	10-24-26, 15-36-39
8, 15, 17	64 + 225 = 289	16-30-34
7, 24, 25	49 + 576 = 625	14-48-50

💡 Pro Tip

If you multiply all three numbers in a Pythagorean triple by the same factor, you get another valid triple! For example, 3-4-5 × 2 = 6-8-10.

Finding the Hypotenuse

When you know the two legs (a and b), use the formula to find the hypotenuse (c):

$c = \sqrt{a^2 + b^2}$

Example 1: Finding the Hypotenuse

Problem:

A right-angled triangle has legs measuring 9 cm and 12 cm. Find the hypotenuse.

Solution:

Apply Pythagoras’ Theorem: $c^2 = a^2 + b^2$
Substitute: $c^2 = 9^2 + 12^2$
Calculate: $c^2 = 81 + 144 = 225$
Square root: $c = \sqrt{225} = 15$
The hypotenuse is 15 cm

Finding a Shorter Side (Leg)

When you know the hypotenuse and one leg, rearrange the formula to find the other leg:

$a = \sqrt{c^2 - b^2}$

Remember: subtract when finding a leg!

Example 2: Finding a Leg

Problem:

The hypotenuse of a right triangle is 17 m and one leg is 15 m. Find the other leg.

Solution:

Rearrange: $a^2 = c^2 - b^2$
Substitute: $a^2 = 17^2 - 15^2$
Calculate: $a^2 = 289 - 225 = 64$
Square root: $a = \sqrt{64} = 8$
The other leg is 8 m

Real-World Application: The Ladder Problem

Pythagoras’ Theorem appears in many real-world contexts. A classic example is the ladder problem.

Example 3: Ladder Against a Wall

Problem:

A 13 m ladder leans against a vertical wall. The base of the ladder is 5 m from the wall. How high up the wall does the ladder reach?

Solution:

The ladder is the hypotenuse (c = 13 m)
The distance from wall is one leg (b = 5 m)
The height is the unknown leg (a = ?)
$a^2 = 13^2 - 5^2 = 169 - 25 = 144$
$a = \sqrt{144} = 12$
The ladder reaches 12 m up the wall

Working with Decimal Answers

Not all problems give neat whole number answers. When the square root doesn’t simplify nicely, use your calculator and round appropriately.

Example 4: Decimal Answer

Problem:

Find the hypotenuse when the legs are 7 cm and 9 cm. Give your answer to 1 decimal place.

$c^2 = 7^2 + 9^2 = 49 + 81 = 130$
$c = \sqrt{130} = 11.4017...$
Answer: 11.4 cm (1 d.p.)

Common Mistakes to Avoid

❌ Mistake 1: Using the Wrong Formula for Finding a Leg

When finding a leg, you must subtract: $a^2 = c^2 - b^2$ . Adding will give you a number larger than the hypotenuse!

❌ Mistake 2: Forgetting to Square Root

The formula gives you $c^2$ , not c. Always take the square root as the final step.

❌ Mistake 3: Misidentifying the Hypotenuse

The hypotenuse is ALWAYS opposite the right angle and is ALWAYS the longest side. Don’t assume based on diagram orientation.

❌ Mistake 4: Using Pythagoras on Non-Right Triangles

This theorem only works for right-angled triangles. For other triangles, you need the Sine or Cosine Rule.

Quick Reference: Which Formula?

You Know	You Want	Formula
Both legs (a and b)	Hypotenuse (c)	$c = \sqrt{a^2 + b^2}$
Hypotenuse + one leg	Other leg	$a = \sqrt{c^2 - b^2}$
Three sides	Is it a right triangle?	Check if $a^2 + b^2 = c^2$

Study Tips for O-Level Success

Memorize common Pythagorean triples — 3-4-5 and 5-12-13 appear frequently.
Always draw and label your triangle — even if a diagram is provided.
Identify the hypotenuse first — this determines which formula to use.
Check your answer makes sense — the hypotenuse must always be the longest side.
Practice word problems — ladders, diagonals, and distances are common contexts.

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Pythagoras' Theorem: The Complete O-Level Math Guide

Pythagoras’ Theorem: The Complete O-Level Guide

What is Pythagoras’ Theorem?

Step 1: Identify the Hypotenuse

The Hypotenuse is…

The Two Legs are…

Pythagorean Triples: Numbers Worth Memorizing

Finding the Hypotenuse

Finding a Shorter Side (Leg)

Real-World Application: The Ladder Problem

Working with Decimal Answers

Common Mistakes to Avoid

Quick Reference: Which Formula?

Study Tips for O-Level Success

Ready to Practice Pythagoras’ Theorem?

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