△ Pythagorean Theorem
Solve any right triangle side and check for Pythagorean triples.
Leave one field blank — it will be calculated.
Result
How to Use This Calculator
This calculator solves all three versions of the Pythagorean theorem: find the hypotenuse from two legs, find a missing leg when you know the hypotenuse and the other leg, or check whether three given lengths form a right triangle. Leave the one side blank that you want to find.
To find the hypotenuse (c), enter both legs (a and b) and leave c blank.
To find a missing leg, enter the hypotenuse and the known leg. Leave the unknown leg blank.
Use the "Check" tab to test whether three measurements actually form a right triangle. This is handy when squaring up a room or framing a wall.
Results show the triangle area and perimeter as bonus information below the main answer.
The Pythagorean Theorem Formula
In this formula, c is always the hypotenuse (the longest side, across from the right angle). The two shorter sides a and b are called legs. The hypotenuse must be longer than either leg, so if your result looks wrong, double-check which side is opposite the right angle. All three sides must be positive numbers.
Worked Examples
Where the Pythagorean Theorem Comes Up
Construction workers use the 3-4-5 rule constantly to check that corners are square. If you measure 3 feet along one wall and 4 feet along the adjacent wall and the diagonal is exactly 5 feet, the corner is a perfect 90 degrees. Carpenters do this when framing walls, tilers use it when laying out floors, and landscapers use it when setting out garden beds.
The theorem is also built into navigation, screen sizing, and physics. Screen sizes are measured diagonally, so if you know the width and height, you can find the screen size. In physics, the magnitude of a 2D vector (like velocity or force) is the hypotenuse of the triangle formed by its horizontal and vertical components. Any time you have two perpendicular quantities and need the combined result, you are using the Pythagorean theorem.
Frequently Asked Questions
What is the Pythagorean theorem?
In a right triangle, the square of the hypotenuse (longest side, opposite the right angle) equals the sum of squares of the other two sides: a² + b² = c².
How do I find the hypotenuse?
c = √(a² + b²). Example: a=3, b=4 → c = √(9+16) = √25 = 5.
How do I find a missing leg?
a = √(c² − b²) or b = √(c² − a²). You need the hypotenuse and one leg.
What are Pythagorean triples?
Sets of three integers that satisfy a²+b²=c². Common ones: (3,4,5), (5,12,13), (8,15,17), (7,24,25), (9,40,41). Multiples also work: (6,8,10).
Does the Pythagorean theorem work in 3D?
Yes — the space diagonal of a rectangular box: d = √(l² + w² + h²). First apply in 2D to get the base diagonal, then apply again with the height.