Pythagorean Theorem Calculator

Enter two sides and solve for a, b, or c. The page uses a² + b² = c².

Smart Tips

• Side c is the hypotenuse, opposite the right angle. Sides a and b are the legs.

• To find c, enter both legs. To find a leg, enter the hypotenuse and the other leg.

• The hypotenuse must be longer than each leg. A shorter hypotenuse is rejected.

• This page stays separate from the triangle calculator. Use that page for oblique triangles.

The Formula

a² + b² = c²

a = Leg
One side of the right angle

b = Leg
The other side of the right angle

c = Hypotenuse
The longest side, opposite the right angle

area = ab / 2
Area of the right triangle

Solving for the hypotenuse takes the square root of a² + b². Solving for a leg takes the square root of c² minus the other leg squared. A 3-4-5 triangle has area 6. Other cases, including angles that are not 90°, belong on the triangle calculator.

Key Insights

One missing side

Enter exactly two sides. The page solves the third from a² + b² = c² and will not invent a second answer.

Hypotenuse check

If you solve for a leg, the hypotenuse has to be strictly longer than the known leg. Otherwise the value under the square root is not positive.

3-4-5 and 5-12-13

Those are exact integer triples. Legs 3 and 4 give hypotenuse 5. Legs 5 and 12 give hypotenuse 13.

A separate page

The query is the formula name, so this page stays on a² + b² = c². The triangle calculator links here and handles the other cases.

Frequently Asked Questions

Side c. It is opposite the right angle. Sides a and b are the legs that meet at the right angle. The hypotenuse is the longest side.

Choose leg a or leg b, then enter the hypotenuse and the other leg. The missing leg is the square root of c² minus the known leg squared. The hypotenuse must be longer than that leg.

Legs 3 and 4 give hypotenuse √(9 + 16) = √25 = 5. The area is 3 × 4 / 2 = 6. The same triple works in reverse: leg 4 and hypotenuse 5 give the other leg 3.

Every side must be greater than zero. If the hypotenuse is not longer than the other side, c² − b² is not positive, so there is no real leg. A zero leg is not a triangle.

Use this page only for a right triangle and the Pythagorean theorem. Use the triangle calculator for SSS, SAS, ASA, AAS, or a right triangle when you also want every angle spelled out from those cases.

In a right triangle, the sum of the squares of the legs equals the square of the hypotenuse, as stated in OpenStax Prealgebra 2e, section 9.3. The missing side is the square root of that sum or difference. No outside data feed is used.

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Missing side

Enter values to calculate

Other sides

Area

Note:The right angle is between sides a and b, so c is the hypotenuse. This page only solves a² + b² = c². Other triangle cases are on the triangle calculator.