Pythagorean Theorem Calculator

In a right triangle, a^2 + b^2 = c^2, so legs of 3 and 4 give a hypotenuse of √(9 + 16) = 5, and a missing leg is √(c^2 − b^2). The calculator also gives the area, perimeter, both acute angles (36.87° and 53.13° here) and exact radical answers such as 5√2.

By the Calculators With Steps team · Updated November 10, 2026 · How we check our math

How to use the Pythagorean theorem calculator

Choose which side you want to find: hypotenuse c or one of the legs, a or b.
Enter the two sides you know. Decimals are fine.
Optionally pick a unit so the answer reads “8 ft” or “15 in”.

The calculator checks that the hypotenuse is longer than the known leg. If it is not, the numbers cannot form a right triangle and you will see a message instead of a wrong answer.

The Pythagorean theorem formula

For a right triangle with legs a and b and hypotenuse c:

a² + b² = c²

Rearranged for whichever side is missing:

Hypotenuse: c = √(a² + b²)
Leg a: a = √(c² − b²)
Leg b: b = √(c² − a²)

The calculator also reports the area (a × b ÷ 2), the perimeter (a + b + c), the height from the right angle to the hypotenuse (a × b ÷ c) and both acute angles, found with arctan(a ÷ b).

Worked examples

Known sides	Working	Missing side	Area	Angles
a = 3, b = 4	√(9 + 16) = √25	c = 5	6	36.87°, 53.13°
a = 5, c = 13	√(169 − 25) = √144	b = 12	30	22.62°, 67.38°
b = 6 ft, c = 10 ft	√(100 − 36) = √64	a = 8 ft	24 ft²	53.13°, 36.87°
a = 1, b = 1	√(1 + 1) = √2	c ≈ 1.414214	0.5	45°, 45°
a = 5, b = 5	√50 = 5√2	c ≈ 7.071068	12.5	45°, 45°

Ladder problem. A 10 ft ladder leans against a wall with its foot 6 ft from the wall. How high does it reach? The ladder is the hypotenuse, so the height is √(10² − 6²) = √64 = 8 ft.

Diagonal of a rectangle. A 9 in × 12 in photo frame has a diagonal of √(81 + 144) = √225 = 15 in. The same idea gives the distance across a rectangular field or room. For the distance between two points on a graph, the slope calculator applies this formula to the coordinates.

Why a² + b² = c² works

One of the oldest proofs needs only areas. Draw a large square with sides of length a + b and fit four copies of the right triangle inside it, one in each corner, so that their hypotenuses form a tilted square in the middle. The large square’s area is (a + b)². The four triangles cover 4 × ½ab = 2ab, and the tilted square covers c². So (a + b)² = 2ab + c². Expanding the left side gives a² + 2ab + b², and taking 2ab from both sides leaves a² + b² = c².

Check it with the 3-4-5 triangle: the large square is 7 × 7 = 49, the four triangles cover 4 × 6 = 24, and the middle square is 5 × 5 = 25. Indeed 24 + 25 = 49.

Pythagorean triples

These primitive triples have no common factor. Multiply any row by a whole number to get more triples (3-4-5 becomes 6-8-10, 9-12-15 and so on).

a	b	c
3	4	5
5	12	13
8	15	17
7	24	25
20	21	29
12	35	37
9	40	41
28	45	53
11	60	61
33	56	65

When all three sides you enter are whole numbers that fit the theorem, the calculator tells you whether it is a primitive triple or a multiple of one.

Common mistakes

Adding the sides instead of their squares. 3 + 4 is 7, but the hypotenuse is 5. Square first, add, then take the root.
Using the hypotenuse as a leg. The hypotenuse is always the longest side and sits opposite the right angle. When finding a leg, subtract: c² − a², not c² + a².
Forgetting the square root. If a² + b² = 25, the side is 5, not 25.
Mixing units. Convert everything to the same unit before you start. Feet and inches together give a wrong answer.
Applying it to a triangle without a right angle. The theorem only holds when one angle is exactly 90°.

Once you know all three sides, the area calculator can confirm the area with Heron’s formula, and the square root calculator shows how answers like √50 simplify to 5√2.

Pythagorean Theorem Calculator: frequently asked questions

What is the Pythagorean theorem formula?

In a right triangle, a² + b² = c², where a and b are the two legs and c is the hypotenuse, the side opposite the right angle. With legs 3 and 4: 3² + 4² = 9 + 16 = 25, so c = √25 = 5.

How do you find the hypotenuse?

Square both legs, add them and take the square root: c = √(a² + b²). With legs 5 and 7, c = √(25 + 49) = √74, which is about 8.602325.

How do you find a missing leg?

Subtract the square of the known leg from the square of the hypotenuse, then take the square root: b = √(c² − a²). With c = 13 and a = 5, b = √(169 − 25) = √144 = 12.

Can I use the Pythagorean theorem if I only know the hypotenuse?

Not on its own. You need two sides, or one side plus an angle, which is a trigonometry problem. The exception is an isosceles right triangle, where both legs are equal: a hypotenuse of 10 gives legs of 10 ÷ √2, about 7.071068 each.

How do I find the angles of a right triangle from its sides?

Use the inverse tangent. The angle opposite leg a is arctan(a ÷ b), and the other acute angle is 90° minus that. For a 3-4-5 triangle the angles are 36.87° and 53.13°.

What is a Pythagorean triple?

A set of three whole numbers that fit a² + b² = c², such as 3-4-5, 5-12-13 and 8-15-17. Any multiple of a triple also works, so 6-8-10 and 9-12-15 are triples too.

Does the Pythagorean theorem work for every triangle?

No, only for right triangles. For other triangles use the law of cosines, c² = a² + b² − 2ab × cos(C), which reduces to the Pythagorean theorem when angle C is 90°.

How do builders use the 3-4-5 rule?

To check that a corner is square, measure 6 ft along one side and 8 ft along the other. If the diagonal between those marks is exactly 10 ft, the corner is a true 90°, because 6² + 8² = 100 = 10².