The Pythagorean Theorem Explained: a² + b² = c²

Q: Does Pythagoras work for all triangles?

No, only right-angled triangles. For other triangles, a^2 + b^2 is bigger or smaller than c^2 depending on whether the angle opposite c is acute or obtuse.

What Pythagoras' theorem says, a picture of why it is true, how to find a missing side (long or short), how to spot a right angle, and Pythagorean triples worth knowing.

3 min read Supports 4 lessons

Draw a square on each side of a right-angled triangle. The two smaller squares together have exactly the same area as the big one on the longest side. Change the legs below and watch the areas.

Interactive

$3^2 + 4^2 = 9 + 16 = 25 = c^2 \;\Rightarrow\; c = 5$

Leg a3

Leg b4

Pythagorean theorem

a^2 + b^2 = c^2

$c$ is the hypotenuse: the longest side, opposite the right angle.

Watch out

The theorem only works for right-angled triangles, and $c$ must be the hypotenuse.

Why it is true

Arrange four copies of the triangle inside a big square of side $a + b$ , with their hypotenuses forming a tilted square in the middle. The big square's area can be counted two ways:

$(a + b)^2$
Area of the big square.

1/4

Finding the hypotenuse

Worked example

The legs of a right-angled triangle are $6$ and $8$ . Find the hypotenuse.

1
Square and add.
$c^2 = 6^2 + 8^2 = 100$

Finding a shorter side

To find a leg, subtract instead: $a^2 = c^2 - b^2$ .

Worked example

A ladder $13$ m long reaches $12$ m up a wall. How far is its foot from the wall?

1
The ladder is the hypotenuse.
$a^2 + 12^2 = 13^2$

Tip

Sense check: the hypotenuse must be the longest side. If your "hypotenuse" comes out shorter than a leg, you added when you should have subtracted, or the other way round.

Testing for a right angle

The theorem works backwards too. If the sides satisfy $a^2 + b^2 = c^2$ , the triangle has a right angle. Builders use a $3$ – $4$ – $5$ rope triangle to set out square corners.

Pythagorean triples

Whole-number solutions are called Pythagorean triples. The most useful are $(3, 4, 5)$ , $(5, 12, 13)$ , $(8, 15, 17)$ and $(7, 24, 25)$ . Any multiple of a triple, such as $(6, 8, 10)$ , is a triple too.

Key idea

Pythagoras reappears all over the course: it is the distance formula on a coordinate grid, the equation of a circle, and the identity $\sin^2\theta + \cos^2\theta = 1$ .

Common questions

Which side is the hypotenuse?: The side opposite the right angle. It is always the longest side of a right-angled triangle.
Does Pythagoras work for all triangles?: No, only right-angled triangles. For other triangles, $a^2 + b^2$ is bigger or smaller than $c^2$ depending on whether the angle opposite $c$ is acute or obtuse.

Practise pythagorean theorem

The Pythagorean Theorem lesson has worked examples and unlimited practice that adapts to you.

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Common questions

Which side is the hypotenuse?

The side opposite the right angle. It is always the longest side of a right-angled triangle.

Does Pythagoras work for all triangles?

No, only right-angled triangles. For other triangles, $a^2 + b^2$ is bigger or smaller than $c^2$ depending on whether the angle opposite $c$ is acute or obtuse.