Pythagoras' theorem – squares on the sides, proofs and applications

What it shows

In a right-angled triangle the square on the hypotenuse equals the sum of the squares on the other two sides: c² = a² + b². The converse is also true: if a² + b² = c², the angle opposite c is 90°. If a² + b² is greater than c² that angle is acute; if it is smaller, the angle is obtuse. The theorem can be proved by rearranging four copies of the triangle inside a square, or with the similar triangles made by the altitude to the hypotenuse. It gives the distance between two points and the diagonal of a cuboid, d = √(l² + w² + h²).

How to use

On Explore, drag A, B and C or press a preset such as 3–4–5, Acute angle C or Obtuse angle C; keep Snap to grid on for exact whole-number areas. On Proofs, choose a Proof and press Play or drag Progress. On Find a side, choose what to Find, move the sliders and press New question to practise. On Distance, drag P and Q. On Cuboid, set l, w and h and drag the picture to rotate it.

Parameters you can change

• Screen Explore, Proofs, Find a side, Distance, Cuboid
• Snap vertices to grid points (Explore)
• Proof Rearrangement, Similar triangles
• Side a (Proofs, Find a side) 1–15 cm
• Side b (Proofs, Find a side) 1–15 cm
• Side to find (Find a side) Hypotenuse, Shorter side
• Length l (Cuboid) 1–12 cm
• Width w (Cuboid) 1–12 cm
• Height h (Cuboid) 1–12 cm

Questions to explore

1. Why do the two smaller squares fill the largest square exactly only when angle C is 90°?
2. How can you use a² + b² and c² to tell whether a triangle is acute, right-angled or obtuse?
3. Why do you need Pythagoras' theorem twice to find the space diagonal of a cuboid?