Pythagoras: How to Use the Theorem (A Brazilian Classroom Favourite)

The theorem in one line

In any right-angled triangle — one with a 90° angle — the square of the hypotenuse equals the sum of the squares of the two legs.

a² + b² = c². c is the hypotenuse (the long side opposite the right angle); a and b are the legs.

If you just need the number, use our Pythagoras calculadora. If you want the idea, read on.

Why it works

The neatest proof fits four copies of the triangle inside a square of side (a + b). What's left in the middle is a square of side c. Counting areas gives you a² + b² = c² directly. Worth drawing once in your life — the "why" then sticks forever.

Classic — legs 3 and 4

3² + 4² = 9 + 16 = 25. √25 = 5.

The 3–4–5 triple is the most famous set of integers satisfying the theorem. Others: 5–12–13, 8–15–17, 7–24–25.

Builders use 3–4–5 to square corners: measure 3m along one line, 4m along the other, and check the diagonal is exactly 5m.

Finding a leg from the hypotenuse

Rearrange: a² = c² − b².

Example: a 5m ladder leans against a wall with its base 3m out. How high up does it touch?

a² = 25 − 9 = 16. a = 4m.

Distance between two points

The Cartesian distance formula is Pythagoras in disguise.

d = √((x₂ − x₁)² + (y₂ − y₁)²).

The x-difference and y-difference are legs of a right triangle; d is the hypotenuse.

Real-world — a 55" TV

"55 inches" is the diagonal. For a 16:9 panel, width and height scale by a factor k so L = 16k, H = 9k.

(16k)² + (9k)² = 55² → 337k² = 3025 → k ≈ 2.997.

Width ≈ 48" (121.9 cm). Height ≈ 27" (68.6 cm). Now you know if it fits the alcove.

Real-world — Part M ramp

A 1:15 accessibility ramp running 6m horizontally rises 0.4m, and has a sloped length of √(6² + 0.4²) ≈ 6.013m. Handrail ordering isn't guesswork once you know this.

When not to use Pythagoras

Only works on right-angled triangles. For any triangle, use the cosine rule: c² = a² + b² − 2ab cos(C). For mixed sides-and-angles, use the sine rule.

Triples worth memorising

• 3 – 4 – 5 (builder's classic)
• 5 – 12 – 13
• 8 – 15 – 17
• 7 – 24 – 25
• 9 – 40 – 41
• Plus all their integer multiples: 6–8–10, 10–24–26, and so on.