Pythagorean Theorem Calculator

What this calculator does

So, you have a right triangle and you know two of its measurements, and you want a third. This tool applies the Pythagorean theorem to find it. Depending on what you pick, it works out a missing leg, the hypotenuse, or the area of the triangle.

A dropdown chooses what to calculate for, and you fill in the two values it needs.

How to use it

1. Choose what to calculate for: leg a, leg b, the hypotenuse, or the area.
2. Enter the two known values it asks for.
3. Press Calculate.

To find the hypotenuse or area, you supply the two legs. To find a missing leg, you supply the other leg and the hypotenuse.

The theorem: a squared plus b squared

The Pythagorean theorem is about right triangles, the ones with a 90 degree corner. It says that the square of the hypotenuse, the long side opposite the right angle, equals the sum of the squares of the other two sides, the legs. Written with legs a and b and hypotenuse c:

a squared + b squared = c squared

So to find the hypotenuse from the two legs, the tool squares each leg, adds them, and takes the square root of the total. Two legs of 3 and 4 give 9 plus 16, which is 25, whose square root is 5.

Finding a leg instead of the hypotenuse

The same equation runs backwards when the missing side is a leg. If you know the hypotenuse and one leg, you rearrange to get the other leg: square the hypotenuse, subtract the square of the known leg, and take the square root of what remains. The tool does exactly this when you ask for leg a or leg b. It is worth noticing that the hypotenuse is always the largest side, so a leg is found by subtracting squares, while the hypotenuse is found by adding them. Getting that direction right is the whole trick.

The area option

The tool also offers the area, which is a different sort of calculation from the theorem itself. In a right triangle the two legs meet at the right angle, so they serve directly as the base and the height. The area is therefore half the product of the two legs. That is why the area option asks for both legs: it multiplies them together and halves the result, no square roots involved. It is a handy extra, since the legs you would enter to find the hypotenuse are exactly the ones needed for the area.

A worked example

Enter legs of 3 and 4 and calculate for the hypotenuse: 3 squared is 9, 4 squared is 16, together 25, and the square root is 5. Now switch to leg a, entering the other leg 4 and the hypotenuse 5: 5 squared minus 4 squared is 25 minus 16, which is 9, whose square root is 3, the leg you started with. Ask for the area with legs 3 and 4 and you get half of 3 times 4, which is 6.

A very old result

This relationship is one of the oldest in mathematics. It appears as Proposition 47 in the first book of Euclid's Elements, written around 300 BC, where it is given a careful geometric proof. The name honours the ancient Greek Pythagoras, though the relationship was known and used far earlier: Babylonian clay tablets from more than a thousand years before him show right-triangle calculations built on it. It has since been proved in a remarkable number of different ways, more perhaps than any other theorem in mathematics.

Questions people ask

What does the Pythagorean theorem say?

In a right triangle, the square of the hypotenuse equals the sum of the squares of the two legs, written a squared plus b squared equals c squared.

How do I find a leg rather than the hypotenuse?

Square the hypotenuse, subtract the square of the known leg, and take the square root. The tool does this when you calculate for leg a or leg b.

Why does the area use the two legs?

Because the legs meet at the right angle, so they act as base and height. The area is half their product.

Does it work for any triangle?

No, only right triangles. For triangles without a right angle, the law of cosines is the generalization to use.

How does it relate to the special right triangle tool?

The 30-60-90 triangle calculator handles that particular right triangle, where a fixed side ratio lets one value give everything. This tool works for any right triangle from two of its sides.

References

On the Pythagorean theorem. In a right triangle the square on the hypotenuse equals the sum of the squares on the other two sides, a result recorded by Euclid and generalized by the law of cosines.

1. Euclid, Elements, Book I, Proposition 47, D. E. Joyce's online edition, Clark University, the classical statement and proof of the theorem.
2. Eric W. Weisstein, "Law of Cosines," from MathWorld, a Wolfram resource, on the generalization that reduces to the Pythagorean theorem at a right angle.