In any right-angled triangle, the square of the hypotenuse (the longest side, opposite the right angle) equals the sum of the squares of the other two sides.
where \(c\) is the hypotenuse and \(a\), \(b\) are the two shorter sides (legs).
Rearranging to find a leg: \(a = \sqrt{c^2 - b^2}\).
The hypotenuse is always:
- Opposite the right angle (marked with a small square)
- The longest side in the triangle
A ladder leans against a wall. The base is 1.8 m from the wall and reaches 4.2 m up the wall. How long is the ladder?
A rectangular paddock is 120 m long and has a diagonal of 150 m. Find the width.
Pythagoras’ theorem extends to 3D problems by applying it in stages.
Diagonal of a rectangular box with dimensions \(l\), \(w\), \(h\):
Worked Example: A box is 6 cm × 4 cm × 3 cm. Find the space diagonal.
Alternatively, find the base diagonal first: \(\sqrt{36+16} = \sqrt{52}\), then apply Pythagoras again: \(\sqrt{52 + 9} = \sqrt{61}\).
Common whole-number solutions worth memorising:
| \((a, b, c)\) | Multiples |
|---|---|
| (3, 4, 5) | (6,8,10), (9,12,15) |
| (5, 12, 13) | (10,24,26) |
| (8, 15, 17) | — |
Recognising triples can save calculation time.
REMEMBER: Pythagoras’ theorem only applies to right-angled triangles. Always confirm the right angle exists before applying the formula.
EXAM TIP: In 3D problems, draw a 2D cross-section showing the right angle. Label all known lengths before applying the theorem. Never try to visualise 3D geometry without a diagram.
