The statement of Pythagoras’ theorem

The fundamental relationship

In a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

If ABC is a right-angled triangle with the right angle at A, and BC is the hypotenuse, then: BC² = AB² + AC²

Numerical Example

Let ABC be a right-angled triangle with right angle at A, where AB = 3 cm and AC = 4 cm. Let’s calculate BC: BC² = AB² + AC² = 3² + 4² = 9 + 16 = 25 BC = √25 = 5 cm

This (3, 4, 5) triangle is very well known and is often used as a reference.

3-step method

1. Identify the right angle and therefore the hypotenuse
2. Write the equation: (hypotenuse)² = (side 1)² + (side 2)²
3. Substitute the known values and calculate the missing length

Summary table

Element	Role
BC	hypotenuse, side opposite the right-angled corner
AB and AC	sides of the right-angled corner
BC² = AB² + AC²	Pythagorean theorem

Pitfall to avoid

The formula can only be written in one specific way: it is always (hypotenuse)^2 that appears ALONE on one side of the equation, and the sum of the other two squares on the other side. Writing AB² = BC² + AC² would be a serious mistake, as AB is not the hypotenuse here. Always check which side is on its own before applying the formula.