Pythagorean Theorem Calculator

What is the Pythagorean theorem?

The Pythagorean theorem states that in any right triangle, the square of the hypotenuse equals the sum of the squares of the two legs: a² + b² = c². Knowing any two sides lets you calculate the third.

How this calculator works

This right triangle calculator solves for any one side when you know the other two:

• Pick what you need from the dropdown: find hypotenuse c, leg a, or leg b.
• Enter the two known sides — any consistent unit works, metric or imperial.
• The missing side appears instantly, with the full formula written out step by step.
• You also get the triangle’s area and a labeled diagram showing which side is which.
• Enter a hypotenuse shorter than a leg and you’ll get an error — that triangle can’t exist.

How to use the result

1. Checking homework: compare each displayed step against your own working — a basic calculator is all you need to verify the arithmetic, but teachers grade the method, not just the final number.
2. Squaring a frame, deck, or wall: measure 3 units along one edge and 4 along the other; if the diagonal isn’t exactly 5, the corner isn’t square. Adjust until it matches.
3. Cutting material: round the result up to the next 1/16 inch or millimeter, then allow for the saw kerf before you cut stock.
4. Estimating sheet goods: use the displayed area to work out how much plywood, roofing, or fabric the triangle will cover.
5. Sanity check: the hypotenuse must come out longer than either leg — if it doesn’t, one of your inputs was wrong.

For quick layout work, a few whole-number triples are worth memorizing:

• 3-4-5 (and any multiple: 6-8-10, 9-12-15…)
• 5-12-13
• 8-15-17
• 7-24-25

How does a Pythagorean theorem calculator find the hypotenuse?

It applies the formula directly: square both legs, add the results, and take the square root — c = √(a² + b²). With legs of 3 and 4, that’s √(9 + 16) = √25 = 5. Solving for a leg runs the same arithmetic in reverse: subtract instead of add, so a = √(c² − b²). Because it handles all three sides, it does more than a single-purpose hypotenuse calculator — and it prints each line of the working, which is the difference between a decent Pythagoras calculator and a bare answer box.

Common mistakes

• Labeling the wrong side c. The hypotenuse is always opposite the right angle and always the longest side — that’s why entering a c smaller than a leg gets rejected here.
• Adding instead of squaring. 3 + 4 = 7, but the hypotenuse is 5. Square first, then take the square root of the total.
• Forgetting the final square root. If a² + b² = 25, the side is 5, not 25 — the most common homework error.
• Mixing units. Convert everything to one unit first — 1 foot and 6 inches should both become inches.
• Applying it to non-right triangles. The theorem only holds with an exact 90° angle; other triangles need the law of cosines.

The same sides feed plenty of follow-up jobs: roof pitch is rise over run, and if you need that ratio expressed as a percent or a grade, a percentage calculator converts it in one step.