Pythagorean Theorem Calculator: Find Any Side

How to use the calculator

Choose what to solve for: the hypotenuse c (the longest side, opposite the right angle) or a leg a or b (the two sides that form the right angle).
Enter the two sides you know. Use any length unit, as long as both sides use the same one; the answer comes out in that unit.
Read the missing side and the formula line, which shows the numbers substituted into the theorem.

When you solve for a leg, the hypotenuse must be longer than the leg you enter, otherwise no right triangle exists and the calculator shows an error.

The theorem and the formulas

In a right triangle the square of the hypotenuse equals the sum of the squares of the legs: a² + b² = c². Rearranged for each unknown:

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

Once all three sides are known, the calculator adds:

Angle A (opposite leg a) = arctan(a ÷ b), and angle B = 90° − A. The third angle is the right angle.
Area = a × b ÷ 2 and perimeter = a + b + c.
Height to the hypotenuse = a × b ÷ c, the perpendicular distance from the right-angle corner to side c.

A Pythagorean triple is reported when all three sides come out as whole numbers. It's primitive when the sides share no common factor (3-4-5, 5-12-13, 8-15-17) and a multiple otherwise. Decimal sides such as 1.5-2-2.5 are a scaled 3-4-5 but aren't reported as a triple.

The theorem also works in reverse: if a² + b² = c² for three side lengths, the angle between a and b is a right angle. For triangles without a right angle, use the triangle calculator, which applies the law of cosines.

Worked examples

1. Squaring a deck corner with 3-4-5

Mark 6 ft along one rim joist and 8 ft along the other. The diagonal between the marks should be: c = √(6² + 8²) = √100 = 10 ft. The calculator labels it 6-8-10 = 2 × (3-4-5). If your tape reads more than 10 ft the corner is open past 90°; less, and it's closed. Larger layouts give a more precise check:

Legs	Diagonal	Triple
3 ft and 4 ft	5 ft	3-4-5 (primitive)
9 ft and 12 ft	15 ft	3 × (3-4-5)
12 ft and 16 ft	20 ft	4 × (3-4-5)

2. How high does a 12-foot ladder reach?

OSHA's construction ladder rule (29 CFR 1926.1053) asks that a non-self-supporting ladder be set so the horizontal distance from the top support to its foot is about one-quarter of its working length. For a 12-ft working length that's 3 ft out. Solve for the leg: a = √(12² − 3²) = √135 = 11.618950 ft up the wall, at an angle of 75.522488° to the ground.

3. Screen diagonal

A screen that measures 48 by 27 inches has a diagonal of √3033 = 55.072679 in, about 55 inches. Its area is 648 in².

Reading the results

The hypotenuse is always the longest side, so a result for c that's shorter than a leg means the inputs were entered in the wrong fields. The two acute angles always add up to 90°. Results are rounded for display; when a side is irrational, such as √2 = 1.414214 for legs of 1 and 1, keep the radical form if your class asks for an exact answer.

Frequently asked questions

What is the Pythagorean theorem?

For any right triangle with legs a and b and hypotenuse c, a² + b² = c². With legs 3 and 4: 9 + 16 = 25, so the hypotenuse is 5.

How do I find a missing leg?

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

Does it work for triangles that aren't right triangles?

No. The theorem only holds when one angle is exactly 90°. For any other triangle, the triangle calculator solves it from three sides, two sides and an angle, or two angles and a side.

Why does the 3-4-5 method square a corner?

Because the theorem works in reverse: if the sides measure 3, 4 and 5 (or any multiple), 3² + 4² = 5² and the angle between the two shorter sides must be 90°. That's why a tape measure is all you need to check a corner.

What is a Pythagorean triple?

Three positive whole numbers that satisfy a² + b² = c². Primitive triples share no common factor; multiplying one by any whole number gives another triple, like 9-12-15 from 3-4-5.

Why do I get an error when solving for a leg?

The hypotenuse you entered is equal to or shorter than the known leg. Since c must be the longest side, c² − a² would be zero or negative and has no real square root.

Sources

Results are computed in floating-point arithmetic and rounded for display. For ladder setup and other safety-critical work, follow the manufacturer's instructions and the rules that apply to you.