Finding a Shorter Side with Pythagoras
Grade 4Rearranging the formula
When you know the hypotenuse and one shorter side, you need to rearrange Pythagoras' theorem to find the missing shorter side.
a² = c² − b²
Subtract the square of the known shorter side from the square of the hypotenuse, then take the square root.
The common mistake
When finding a shorter side, you subtract — not add. The classic error is to add both squares. Always ask: am I finding the hypotenuse (add) or a shorter side (subtract)?
Worked example
EXAMPLE
A right-angled triangle has hypotenuse 15 cm and one side of 9 cm. Find the third side.
a² = c² − b² = 15² − 9² = 225 − 81 = 144. a = √144 = 12 cm.
Real-world application
EXAMPLE
A ladder of length 10 m leans against a wall. The base of the ladder is 4 m from the wall. How high up the wall does the ladder reach?
Height² = 10² − 4² = 100 − 16 = 84. Height = √84 ≈ 9.17 m (to 3 s.f.).
