Pythagorean Theorem Calculator

Find the missing side of a right triangle using the Pythagorean theorem. Calculate area, perimeter, and angles.

Inputs

Solve For

Side A

Side B

Tab to move between fields

Results

Result

Hypotenuse5

Triangle Area6

Perimeter12

Angle A36.87°

Angle B53.13°

The Pythagorean theorem

1.Square both legs and add them
$a^2 + b^2$
The theorem is a statement about areas, not lengths: the square built on the hypotenuse has exactly the combined area of the squares built on the two legs.
2.Take the square root to get back to a length
$c = sqrt(a^2 + b^2)$
The previous step is an area. The square root converts it back into the distance you actually wanted.

Euclid's proof arranges the three squares around the triangle and shows the two smaller ones can be cut and rearranged to fill the largest. Babylonian tablet Plimpton 322 lists Pythagorean triples roughly a thousand years earlier, so the relationship was in use long before it was proved.

Euclid, Elements, Book I, Proposition 47·c. 300 BCE

How to Use This Calculator

Choose the mode: find the hypotenuse (c) from legs a and b, or find a missing leg from the hypotenuse and one leg.
The Pythagorean theorem states: a² + b² = c², where c is always the hypotenuse (the longest side, opposite the right angle).
Enter the two known sides; the third is derived by solving the equation algebraically.
The calculator also reports the triangle's area (½ × a × b) and perimeter (a + b + c).
Angles A and B (opposite sides a and b) are found using inverse trig: angle A = arctan(a/b).
All side lengths must be positive; the hypotenuse must be longer than either leg.

How the result changes with Side B

Side B	Result
1,000	1,000
3,500	3,500
6,500	6,500
9,000	9,000

What each input means

Solve For: Calculation mode to use.
Side A: When finding missing side, this is the hypotenuse.
Side B: Length of side B.

How this is calculated

Worked example, using the default values

Identify Input Parameters
Solve For = 1, Side A = 3, Side B = 4 = 3 input(s) provided
Calculate Result
Result = missingSide
5 = 5
Calculate Hypotenuse
Hypotenuse
5 = 5
Calculate Triangle Area
Triangle Area
6 = 6