Introduction
The Pythagorean theorem says that in a right triangle, the two short sides squared and added together equal the long side squared. We write it as a² + b² = c², where a and b are the legs and c is the hypotenuse. This rule only works for right triangles — triangles that have one 90-degree angle.
This Pythagorean theorem calculator lets you find any missing side of a right triangle in seconds. Enter the two sides you know, and it solves for the third. You can also use it to check if three sides form a right triangle. Every calculation comes with a step-by-step solution, a triangle diagram, and a visual proof that shows how the squares on each side relate to one another. Choose between exact radical answers or rounded decimals, pick your unit label, and try quick presets like the classic 3-4-5 triangle to see how it works.
How to Use Our Pythagorean Theorem Calculator
Enter the side lengths you know, and this calculator will find the missing side of a right triangle. It also gives you the area, perimeter, angles, step-by-step work, and a triangle diagram.
Calculation Mode: Pick what you want to solve. Choose "Find Hypotenuse (c)" to find the longest side, "Find Leg a" or "Find Leg b" to find a shorter side, or "Right Triangle Check" to test if three sides form a right triangle.
Coefficient: Type the number part of each side length. For example, if a side is 5, type 5 here.
Under √ (Radical): If your side length has a square root, type the number under the radical sign. For example, for 3√2, type 3 as the coefficient and 2 here. Leave this blank for whole numbers.
Units: Pick a unit label like cm, ft, or m. This does not convert anything. It just adds the unit name to your results.
Output Format: Choose how you want your answer shown. "Exact (radical)" keeps square roots in the answer. "2 Decimal" and "3 Decimal" round to that many decimal places. "Sig Figs" lets you set how many significant figures to show.
Quick Pick Presets: Click a preset button like 3-4-5 or 5-12-13 to fill in a common right triangle automatically.
Display Options: Use the toggles to show or hide the step-by-step solution, the triangle diagram, and the square-area visual.
Calculate: Press the Calculate button to get your results. You can also press Enter on your keyboard from any input field.
What Is the Pythagorean Theorem?
The Pythagorean Theorem is a math rule that works on right triangles. A right triangle is any triangle that has one 90-degree angle. The theorem says that if you square the two shorter sides (called legs) and add them together, you get the square of the longest side (called the hypotenuse). Written as a formula, it looks like this: a² + b² = c², where a and b are the legs and c is the hypotenuse.
How the Formula Works
Squaring a number means multiplying it by itself. If one leg is 3 and the other leg is 4, you get 3² + 4² = 9 + 16 = 25. The square root of 25 is 5, so the hypotenuse is 5. This set of whole numbers — 3, 4, 5 — is called a Pythagorean triple. Other common triples include 5-12-13, 8-15-17, and 7-24-25.
Finding a Missing Leg
You can also use the theorem to find a missing leg. Rearrange the formula to a = √(c² − b²). Subtract the square of the known leg from the square of the hypotenuse, then take the square root. The hypotenuse must always be longer than either leg, or the math will not work.
Checking if a Triangle Is a Right Triangle
If you know all three sides, plug them into the formula. If a² + b² equals c² (using the longest side as c), the triangle is a right triangle. If the two sides do not match, it is not a right triangle. For non-right triangles, you can use the Law of Cosines to work with any triangle shape.
Where the Pythagorean Theorem Is Used
This theorem shows up everywhere in real life. Builders use it to make sure corners are square when framing walls or cutting rafters. Engineers use it to find distances. It is used in navigation, architecture, video games, and physics. Any time you need to find a straight-line distance between two points, the Pythagorean Theorem can help.