Pythagorean Theorem Calculator
Enter any two sides of a right triangle and leave the unknown side blank. The calculator will find the missing side and show every step of the work.
Solution
How the Pythagorean Theorem Works
In any right triangle, the square of the hypotenuse equals the sum of the squares of the two legs:
$$a^2 + b^2 = c^2$$The hypotenuse is always the side opposite the right angle — it is the longest side.
Solving for Each Side
Rearranging the formula lets you solve for whichever side is missing:
| Unknown | Formula |
|---|---|
| Hypotenuse c | $$c = \sqrt{a^2 + b^2}$$ |
| Leg a | $$a = \sqrt{c^2 - b^2}$$ |
| Leg b | $$b = \sqrt{c^2 - a^2}$$ |
Worked Example — Finding the Hypotenuse
A right triangle has legs $a = 6$ and $b = 8$. Find the hypotenuse $c$.
$$c^2 = a^2 + b^2$$ $$c^2 = 6^2 + 8^2$$ $$c^2 = 36 + 64 = 100$$ $$c = \sqrt{100} = 10$$Worked Example — Finding a Leg
A right triangle has hypotenuse $c = 13$ and leg $b = 5$. Find leg $a$.
$$a^2 = c^2 - b^2$$ $$a^2 = 13^2 - 5^2$$ $$a^2 = 169 - 25 = 144$$ $$a = \sqrt{144} = 12$$Common Pythagorean Triples
A Pythagorean triple is a set of three whole numbers that satisfy $a^2 + b^2 = c^2$. Memorizing these can save time:
- 3, 4, 5 — and multiples: 6-8-10, 9-12-15, …
- 5, 12, 13
- 8, 15, 17
- 7, 24, 25
Frequently Asked Questions
Does the Pythagorean theorem work for any triangle?
No — only right triangles (triangles with a 90° angle). For non-right triangles, use the Law of Cosines.
What if my answer is not a whole number?
That's perfectly normal. Most triangles produce irrational hypotenuses. The calculator shows a decimal approximation rounded to four decimal places.
How do I know which side is the hypotenuse?
The hypotenuse is always opposite the right angle and is always the longest of the three sides.
