The Pythagorean theorem

The Pythagorean theorem ties together the three sides of a right triangle — and only that kind. It is what lets you compute a length nobody can measure directly: the diagonal of a screen, the height a ladder reaches up a wall, a distance as the crow flies.

Naming the sides

• The legs $a$ and $b$ — the two sides forming the right angle.
• The hypotenuse $c$ — the side opposite the right angle, always the longest.

Before substituting anything, work out which side is the hypotenuse. You can spot it without arithmetic: it is the one that does not touch the right-angle mark.

The theorem

In words: the squares on the two legs add up to the square on the hypotenuse.

Why? The square built on side $a$ has area $a^2$, the square on side $b$ has area $b^2$. The theorem says those two squares together have exactly the area of the square built on the hypotenuse. It is a statement about areas, not about lengths alone — which is why second powers appear everywhere in it.

Finding the hypotenuse

Finding a leg

The theorem works in reverse too. Given the hypotenuse and one leg, the other comes from subtraction:

Pythagorean triples

Some triples of whole numbers satisfy the theorem exactly. They are worth memorising, because problems reach for them most often:

Each one scales: since $3\text{-}4\text{-}5$ works, so do $6\text{-}8\text{-}10$ and $9\text{-}12\text{-}15$.

Outside those cases the answer usually is not a whole number. For legs of $2$ and $3$ we get $c = \sqrt{4 + 9} = \sqrt{13} \approx 3.61$ — and that is a perfectly normal answer. The form $\sqrt{13}$ is exact, $3.61$ is its approximation.

The converse

It also holds the other way round: if the sides of a triangle satisfy $a^2 + b^2 = c^2$, then that triangle is right-angled.

So a right angle can be checked without a protractor — just measure the three sides. That is the basis of the builder's 3-4-5 method: mark $3$ units along one wall, $4$ along the other, and if the distance between the marks is exactly $5$, the corner is square.

An application: the diagonal of a rectangle

The diagonal splits a rectangle into two right triangles whose legs are its sides.

The special case is a square, where both legs are equal:

For a side of $5\,\text{cm}$ the diagonal is $5\sqrt{2} \approx 7.07\,\text{cm}$.

The answer is a length, so it is given in centimetres or metres. If your data is in inches or feet — and screen diagonals are quoted in inches — convert it first with our length converter.

Exercises

Type the answer together with its unit, e.g. 5 cm. Every question is built on a Pythagorean triple, so the answers are whole numbers.

Common mistakes

• Using the theorem in a triangle with no right angle — $a^2 + b^2 = c^2$ holds only in a right triangle.
• Mistaking the hypotenuse for a leg — $c$ is always the side opposite the right angle, and always the longest.
• Adding instead of subtracting when finding a leg — that one is $b = \sqrt{c^2 - a^2}$, not $\sqrt{c^2 + a^2}$.
• Forgetting the square root — $c^2 = 25$ is not the answer yet; the answer is $c = 5$.
• $\sqrt{a^2 + b^2} = a + b$ — the root of a sum is not the sum of the roots; $\sqrt{9+16} = 5$, not $3 + 4 = 7$.
• Expecting a whole number every time — outside the Pythagorean triples the answer is irrational, and that is correct.