Pythagorean Theorem Calculator
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- No sign-up
- Works on phones
- Shows the working
By Yasir Hafeez · Updated
This Pythagorean theorem calculator finds the missing side of a right triangle. Enter any two sides (the legs a and b, or one leg and the hypotenuse c), leave the third blank, and it returns the missing side as an exact simplified square root and a decimal, the two acute angles, the area and the perimeter. It also tells you when the sides form a Pythagorean triple.
It’s free and needs no sign-up.
How to use it
- Enter two of the three sides. c is always the hypotenuse, the longest side, opposite the right angle.
- Leave the unknown side empty.
- Press Solve.
The formula
$$a^2 + b^2 = c^2$$
| You want | 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: the hypotenuse
Legs 3 and 4 (as loaded above): \( c = \sqrt{9 + 16} = \sqrt{25} = 5 \). The angles are about \( 36.87^\circ \) and \( 53.13^\circ \), and the area is \( \frac12(3)(4) = 6 \).
Worked example: a missing leg
A 25 ft ladder reaches 24 ft up a wall. How far is its foot from the wall? \( a = \sqrt{25^2 - 24^2} = \sqrt{625 - 576} = \sqrt{49} = 7 \) ft.
Worked example: an irrational answer
Legs 2 and 5: \( c = \sqrt{4 + 25} = \sqrt{29} \approx 5.385 \). Most answers aren’t whole numbers; the exact form \( \sqrt{29} \) is what teachers usually want.
Pythagorean triples
Whole-number solutions are called Pythagorean triples: (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25), and any multiple of them, such as (6, 8, 10). Spotting one saves calculation.
Tips and common mistakes
- Putting the hypotenuse in the wrong place. \( c \) must be the longest side; \( a^2 + c^2 = b^2 \) only if b is the hypotenuse.
- Adding instead of subtracting when finding a leg.
- Taking the square root too early. \( \sqrt{9 + 16} \neq 3 + 4 \).
- Non-right triangles. The theorem only works with a \( 90^\circ \) angle; otherwise use the law of cosines or the triangle calculator.
Related guides
- The Pythagorean Theorem: Proof and Examples
- Special Right Triangles
- Distance and Midpoint Formula
- Law of Sines and Cosines
Further reading
- Pythagorean theorem (Wikipedia) — many proofs, history and generalizations.
- Pythagorean triple (Wikipedia) — how triples are generated.
FAQ
What is the Pythagorean theorem?
In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: \( a^2 + b^2 = c^2 \).
How do I find the hypotenuse?
\( c = \sqrt{a^2 + b^2} \).
How do I find a missing leg?
\( a = \sqrt{c^2 - b^2} \).
Does it work for any triangle?
Only for right triangles. For others, use the law of cosines.