The Pythagorean theorem and a proof you can draw

TL;DR:

• In a right triangle with legs a, b and hypotenuse c, a² + b² = c².

• A picture of four copies of the triangle inside a big square proves it.

• The converse tells you whether a triangle is right-angled, and the distance formula is the theorem in coordinates.

A rectangular screen is 4 units wide and 3 tall. How long is its diagonal? The Pythagorean theorem answers that: 5. The more interesting question is why, and you can answer it with a drawing and two lines of algebra.

The statement

In a right triangle, the hypotenuse is the side opposite the right angle, and it is the longest. If the other two sides (the legs) have lengths a and b, and the hypotenuse has length c, then a² + b² = c². In words: the square on the hypotenuse has the same area as the two squares on the legs added together.

Worked example: a 3-4-5 right triangle

1. Square the legs: 3² = 9 and 4² = 16.

2. Add them: 9 + 16 = 25.

3. Take the square root: c = √25 = 5.

Check the other way: 5² = 25, which equals 3² + 4². The diagonal is 5 units.

Try to find a missing leg. A ladder 13 units long reaches 12 units up a wall. The foot is a from the wall, so a² + 12² = 13², and a² = 169 − 144 = 25, so a = 5. Always subtract when the unknown is a leg, and add when the unknown is the hypotenuse.

A proof you can draw

Draw a big square with sides a + b. In each corner place a copy of the right triangle, with legs a and b, so that the hypotenuses form a tilted square in the middle. Each side of that tilted square is c.

1. The big square has area (a + b)², which expands to a² + 2ab + b².

2. The four triangles each have area ½ab, together 2ab.

3. The tilted square in the middle has area c².

4. Total area: 2ab + c². Set the two expressions for the big square equal: a² + 2ab + b² = 2ab + c².

5. Subtract 2ab from both sides: a² + b² = c².

Check with numbers, using a = 3, b = 4, c = 5. The big square has area 7² = 49. The triangles total 4 × 6 = 24, and the middle square has area 25. Then 24 + 25 = 49. The drawing and the algebra agree.

The converse: is a triangle right-angled?

The theorem also works in reverse. If the three sides satisfy a² + b² = c², the triangle has a right angle. Test 5, 12, 13: 25 + 144 = 169 = 13², so yes. Test 6, 7, 9: 36 + 49 = 85, but 9² = 81, so no. Because 85 > 81, the angle opposite the longest side is smaller than a right angle. Carpenters have used the 3-4-5 triangle to check square corners this way.

The distance formula is the same idea

The distance between (1, 2) and (7, 10) is the hypotenuse of a triangle whose legs run across and up. Across: 7 − 1 = 6. Up: 10 − 2 = 8. So d² = 6² + 8² = 36 + 64 = 100, and d = 10. In general, d = √((x₂ − x₁)² + (y₂ − y₁)²).

Try one yourself: the shortcut across a field

A path runs 60 m east and then 80 m north. Walking the path is 60 + 80 = 140 m. The straight line across is the hypotenuse: √(60² + 80²) = √(3600 + 6400) = √10000 = 100 m. The shortcut saves 140 − 100 = 40 m.

A screen that is 40 cm wide and 30 cm tall has a diagonal of √(1600 + 900) = √2500 = 50 cm, a 3-4-5 triangle scaled by 10. Spotting scaled copies of 3-4-5, 5-12-13 and 8-15-17 saves a lot of arithmetic.

Where whole-number triples come from

You can generate right triangles with whole sides. Pick two whole numbers m > n and form m² − n², 2mn and m² + n². With m = 2, n = 1, you get 4 − 1 = 3, 2 × 2 × 1 = 4 and 4 + 1 = 5. With m = 3, n = 2, you get 9 − 4 = 5, 2 × 3 × 2 = 12 and 9 + 4 = 13. Check the second: 25 + 144 = 169. The reason it always works is that (m² − n²)² + (2mn)² = m⁴ − 2m²n² + n⁴ + 4m²n² = m⁴ + 2m²n² + n⁴ = (m² + n²)².

How to check your answer

• Hypotenuse is longest. If your c is shorter than a leg, you subtracted when you should have added.

• Triangle inequality. The hypotenuse must be less than a + b. Here 5 < 3 + 4 = 7.

• Plug back. Square all three sides and see if the sum works.

• Units. The answer has the same unit as the sides, not square units.

Common mistakes

• Using the theorem on a triangle with no right angle.

• Writing a + b = c. For 3, 4, 5 this says 7 = 5.

• Forgetting to take the square root at the end.

• Adding the legs' squares when you need a missing leg.

• Squaring the sum instead of summing the squares: (3 + 4)² = 49, but 3² + 4² = 25.

Where you will use it next

Right triangles lead to the ratios in SOH CAH TOA and when to use each ratio and to the unit circle, where x² + y² = 1 is the theorem for a hypotenuse of 1. Triangles of the same shape also scale together, as in similar triangles.

Frequently asked questions

Does it work in three dimensions?

Yes. The diagonal of a box with sides a, b, c satisfies d² = a² + b² + c². For a 1 by 2 by 2 box, d² = 1 + 4 + 4 = 9, so d = 3.

Are there other whole-number triples?

Yes, such as 5, 12, 13 and 8, 15, 17. Check the second: 64 + 225 = 289 = 17².