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Geometry · 12 min

Pythagoras’ theorem

The theorem connects horizontal and vertical distances to a diagonal. It is useful in construction, maps and coordinate geometry.

01Why this matters

The theorem connects horizontal and vertical distances to a diagonal. It is useful in construction, maps and coordinate geometry.

02See it

4 m3 m5 m
The diagonal lies opposite the right angle. Its square is 4² + 3² = 25, so its length is 5 m. Diagram is schematic.
a² = 9b² = 16c² = 25c = 5 m
Read left to right: a² = 9 → b² = 16 → c² = 25 → c = 5 m. Each panel connects one part of the example.

03Understand it

In a right-angled triangle, the square of the longest side equals the sum of the squares of the other two sides.

The longest side, opposite the right angle, is the hypotenuse c. The other sides are a and b. The relationship a² + b² = c² applies only to right-angled triangles in flat geometry. After finding a squared length, take its positive square root to recover the length.

04Guided example

Find the diagonal of a rectangle 3 m wide and 4 m long.

  1. The diagonal forms a right triangle with the rectangle’s sides. Set a = 3 and b = 4.
  2. Calculate c² = 3² + 4² = 9 + 16 = 25.
  3. Take the positive square root: c = √25 = 5 m.

05Try it

Right triangle legs 6 m and 8 m. Hypotenuse?
A little hint

Square, add, then take the square root.

06Practise it

Hypotenuse 13 cm, one leg 5 cm. Other leg?
A little hint

Subtract 5² from 13².

A square has side 3 m. What is its diagonal squared?
A little hint

Add the squares of two equal sides.

07Use it in the real world

A ladder 5 m long with its foot 3 m from a wall reaches 4 m vertically in the idealised right-triangle model. This is a geometry example, not advice on safe ladder placement.

Go a little deeper

The converse is also true: if the side lengths satisfy a² + b² = c² with c longest, the triangle is right-angled.

08Quick check

Right triangle legs 9 and 12 m. Hypotenuse?
A little hint

Calculate √(81 + 144).

09Summary

In a right-angled triangle, the square of the longest side equals the sum of the squares of the other two sides. Take the positive square root: c = √25 = 5 m.

Before moving on, explain the worked example in your own words. Use the hint on any question you want to revisit.

10Key idea

a² + b² = c²

a and b are perpendicular side lengths; c is the hypotenuse, all in the same units.

c = √(a² + b²); a = √(c² − b²)

Find this in the formula library →