Pythagorean Theorem — Formula, Proof & Examples
Key Points At A Glance
- The Pythagorean theorem applies only to right-angled triangles.
- Formula: a² + b² = c², where c is the hypotenuse.
- The hypotenuse is the longest side, opposite the right angle.
- Add the squares to find the hypotenuse; subtract to find a missing leg.
- The converse checks whether a triangle is right-angled.
- The 3-4-5 triangle is the simplest example of the theorem.
The Pythagorean theorem is one of the most famous rules in all of mathematics — and one of the most useful. Builders, designers, game developers and navigators all rely on it every day. The best part? Once you understand the single idea behind it, you can solve a huge range of problems. Let's break it down simply.
What Is the Pythagorean Theorem?
The Pythagorean theorem describes a special relationship in a right-angled triangle (a triangle with one 90° angle).
It states:
In formula form:
- a² + b² = c²
Here:
- c is the hypotenuse — the longest side, always opposite the right angle.
- a and b are the other two sides, called the legs.
Understanding Each Part
Before using the formula, make sure you can spot the parts of the triangle:
- The right angle is the 90° corner (usually marked with a small square).
- The hypotenuse is always opposite that right angle and is the longest side.
- The two legs form the right angle itself.
Worked Example 1 — Finding the Hypotenuse
A right triangle has legs of 3 cm and 4 cm. Find the hypotenuse.
- a² + b² = c²
- 3² + 4² = c²
- 9 + 16 = c²
- 25 = c²
- c = √25 = 5 cm
This famous 3-4-5 triangle is the simplest example of the theorem in action.
Worked Example 2 — Finding a Missing Leg
The hypotenuse is 13 cm and one leg is 5 cm. Find the other leg.
- a² + b² = c²
- 5² + b² = 13²
- 25 + b² = 169
- b² = 169 − 25 = 144
- b = √144 = 12 cm
Notice that to find a leg, you subtract; to find the hypotenuse, you add.
The Converse of the Theorem
The theorem also works in reverse, which is very handy:
So you can use it to check whether a triangle has a right angle. For example, a triangle with sides 6, 8 and 10 is right-angled because 6² + 8² = 36 + 64 = 100 = 10².
Where It Is Used in Real Life
- Construction — checking that walls and corners are perfectly square.
- Navigation and maps — finding the shortest straight-line distance.
- Screens and TVs — the size is the diagonal, found using this theorem.
- Sports and games — calculating distances on a field or grid.
Common Mistakes to Avoid
- Forgetting that c must be the hypotenuse (the longest side).
- Adding when you should subtract to find a missing leg.
- Forgetting to take the square root at the end.
- Using it on a triangle that is not right-angled.
Quick Summary
- The theorem only works for right-angled triangles.
- Formula: a² + b² = c², where c is the hypotenuse.
- Add the squares to find the hypotenuse; subtract to find a leg.
- The converse lets you test whether a triangle is right-angled.
Maths sticks best through practice, so solve two or three triangles by hand right after reading this. When you move on to equations, our Quadratic Equations notes build on the same squaring idea. To revise efficiently before exams, try the methods in How to Study Smart for Exams, and browse all Mathematics notes any time.
Frequently Asked Questions
It says that in a right-angled triangle, the square of the longest side (hypotenuse) equals the sum of the squares of the other two sides — written as a² + b² = c².
The hypotenuse is the longest side of a right-angled triangle. It is always the side opposite the 90° angle.
No. It only works for right-angled triangles — triangles that contain one 90° angle.
Rearrange the formula and subtract: leg² = hypotenuse² − other leg². Then take the square root of the result.
The converse states that if a² + b² = c² is true for a triangle's sides, then the triangle is right-angled. It is used to check for a right angle.
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