Calculate the side of a right-angled triangle

When the unknown is not the hypotenuse

Sometimes, the hypotenuse is known and one of the sides of the right-angled triangle needs to be found. In this case, you need to adapt the formula by isolating the missing square.

The adapted formula

If BC² = AB² + AC², and we are looking for AC (given BC and AB), then: AC² = BC² - AB²

We subtract; we no longer add, because it is one of the sides of the right-angled triangle that is missing, not the hypotenuse.

Step-by-step example

Let ABC be a right-angled triangle with the right angle at A, where BC = 13 cm (hypotenuse) and AB = 5 cm. Let’s calculate AC: AC² = BC² - AB² = 13² - 5² = 169 - 25 = 144 AC = √144 = 12 cm

We find the triple (5, 12, 13), another very common triple.

Comparison of the two situations

What is being sought	Formula to use
The hypotenuse	(hypotenuse)² = (side 1)² + (side 2)²
One side of the right-angled triangle	(unknown side)² = (hypotenuse)² - (known side)²

Common pitfall

The most common mistake is to add instead of subtract when finding one side of the right-angled triangle. Remember: the hypotenuse is always the largest value, so if the result of a subtraction is a negative number, this indicates an error (the wrong side has been identified as the hypotenuse, or the values have been swapped).