In any right-angled triangle the square on the hypotenuse equals the sum of the squares on the other two sides. Use it whenever you need a missing length and no angle is involved — and recognise it hiding inside rectangles, isosceles triangles and 3D solids.
When the unknown is the longest side (opposite the right angle), add the squares: c = √(a² + b²).
When the unknown is one of the shorter sides, subtract: a = √(c² − b²). Getting these two the wrong way round is the most common slip.
Diagonals of rectangles, the height of an isosceles triangle, and space diagonals of cuboids are all Pythagoras in disguise.
A right-angled triangle has legs 5 cm and 12 cm. Find the hypotenuse.
A right-angled triangle has hypotenuse 25 cm and one leg 7 cm. Find the other leg.
A right-angled triangle has legs of 9 cm and 12 cm. Find the hypotenuse.
A right-angled triangle has hypotenuse 26 cm and one leg 24 cm. Find the length of the other leg.
A rectangle measures 8 cm by 15 cm. Calculate the length of a diagonal.
An isosceles triangle has two equal sides of 10 cm and a base of 12 cm. Calculate its perpendicular height.
A right-angled triangle has hypotenuse 20 cm. One of the shorter sides is 7 cm longer than the other. Find the lengths of the two shorter sides.