Pythagorean Theorem Calculator
The right-triangle workhorse: a² + b² = c². Tell the tool which side you're missing — the hypotenuse or one leg — and it returns the length plus the exact steps. Handy for the classic 3-4-5 squaring trick on a job site, or finding a room's diagonal.
What Are You Solving For?
The Pythagorean Theorem vs Other Side-Finding Tools
The theorem only works on a right triangle. If your triangle isn't right-angled, or you need a diagonal through 3D space, a different tool applies. Here's how to pick:
Pick the right formula for the shape you have
| Use this when…Best | Formula | Example |
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Law of cosines becomes the Pythagorean theorem when the angle C is 90° (cos 90° = 0) — the theorem is just its right-angle special case.
How to Find a Missing Side Step by Step
- Confirm a right angle exists. If not, switch to the law of cosines.
- Identify the missing side. The hypotenuse is always the longest side and sits opposite the right angle.
- For the hypotenuse: square both legs, add, take the square root — c = √(a² + b²).
- For a missing leg: square the hypotenuse, subtract the square of the known leg, take the root — a = √(c² − b²).
- Check it makes sense. The hypotenuse must be longer than either leg, and shorter than their sum.
Worked Example: Will a 65-Inch TV Fit a 50-Inch Wide Stand?
TV size is the diagonal. A 65-inch screen in a 16:9 ratio has width ≈ 65 × (16/√337) ≈ 56.7 in and height ≈ 32 in. That already exceeds a 50-inch stand, so the answer is no — but a 55-inch TV (width ≈ 48 in) fits. The relevant calculator step is the reverse problem: if you know the stand is 50 in wide and you want a 16:9 set, the max diagonal is √(50² + 28.1²) ≈ 57.4 in. Use the theorem to size backward from the space you have.
Frequently Asked Questions
What is the Pythagorean theorem?
a² + b² = c² — the hypotenuse squared equals the sum of the legs squared.
How do I find the hypotenuse?
c = √(a² + b²). For 3 and 4, c = 5.
How do I find a missing leg?
a = √(c² − b²). With c = 5, b = 4, a = 3.
How do I check if a triangle is right-angled?
Square the two shorter sides and add; if it equals the longest side squared, it's right-angled.
Where is it used in real life?
Squaring foundations (3-4-5 rule), TV/room diagonals, straight-line distances, navigation.