If you know two sides of a right triangle, the third one is never really a mystery; it just takes one formula. This calculator finds a missing side, the hypotenuse or a leg, once you enter the other two. It works with sides only, so if you are trying to find an angle instead, you will want a different tool, covered further down this page. Otherwise, enter what you know, and the missing length comes right back.
What Is the Pythagorean Theorem?
The Pythagorean theorem describes a fixed relationship between the three sides of any right triangle. It states that a squared plus b squared equals c squared, where c is the hypotenuse, the longest side, sitting directly opposite the right angle, and a and b are the two legs. This relationship only holds when one of the triangle's angles is exactly 90 degrees. Once you know any two of the three sides, the formula gives you the third without needing to measure it directly.
How Does This Calculator Work?
Enter any two known sides, two legs, or one leg and the hypotenuse, and press calculate. The missing side comes back immediately, worked out from the formula above. One thing worth knowing up front: this tool solves for one side only. It does not need an angle to work, and it will not return one either. If your problem involves finding an angle rather than a length, that is a separate calculator, and the difference between the two is explained later on this page.
How Do You Find the Hypotenuse or a Missing Leg by Hand?
Picture a ladder leaning against a house. The ladder itself is the hypotenuse, the height of the house is one leg, and the distance from the wall to the base of the ladder is the other. Say the house is 3 meters tall and the ladder's base sits 4 meters from the wall. The hypotenuse works out to the square root of 3 squared plus 4 squared, which is the square root of 25, or 5 meters. That is the length of the ladder.
Finding a leg instead of the hypotenuse works the same way, just rearranged: a equals the square root of c squared minus b squared. There is also one special case worth knowing. If both legs of a right triangle, an isosceles right triangle, are equal in length, each leg works out to the hypotenuse divided by the square root of 2.
How Do You Check if Three Sides Form a Right Triangle?
Sometimes all three side lengths are already known, and the question is simply whether the triangle they form is a right triangle at all. Sort the three sides so the largest is treated as the hypotenuse, then test whether the sum of the squares of the two shorter sides equals the square of the longest one.
Result of the Check | Type of Triangle |
The sum of the two smaller squares equals the largest square | Right triangle |
The sum of the two smaller squares is less than the largest square | Obtuse triangle |
The sum of the two smaller squares is more than the largest square | Acute triangle |
What Are Pythagorean Triples?
A Pythagorean triple is a set of three whole numbers that satisfy the formula exactly, with no rounding involved. The smallest and most common ones are 3, 4, and 5. Multiplying every number in a known triple by the same value produces another valid triple, which is why 6, 8, and 10 also work.
Triple | Check |
3, 4, 5 | 3² + 4² = 9 + 16 = 25 = 5² |
5, 12, 13 | 5² + 12² = 25 + 144 = 169 = 13² |
8, 15, 17 | 8² + 15² = 64 + 225 = 289 = 17² |
7, 24, 25 | 7² + 24² = 49 + 576 = 625 = 25² |
What Common Mistakes Cause Wrong Answers?
A small number of mix-ups explain most incorrect results.
Mistake | What Actually Happens |
Mixing up which side is the hypotenuse | Treating a leg as the longest side instead of the side opposite the right angle |
Squaring or adding incorrectly | Skipping a step when squaring each side or adding the two smaller squares together |
Mixing units in the same calculation | Entering one side in inches and another in centimeters without converting first |
Where Does This Fit Among Other Triangle and Geometry Calculators?
The Pythagorean theorem is just one piece of a wider set of triangle and geometry tools. Depending on whether you need an angle, a distance between two points, or a full solve rather than just one missing side, a different calculator on this site might be the faster route.
How Is a Right Triangle Calculator Different?
This page only finds a missing side once two other sides are known. A right triangle calculator goes further, solving angles, altitude, area, and perimeter from any two known values, whether those are two sides or a side and an angle.
How Do You Use This Formula to Find the Distance Between Two Points?
The distance between two points on a coordinate grid uses this exact same formula, treating the horizontal and vertical gap between the points as the two legs of a right triangle. For that specific case, with coordinates instead of a triangle drawn on paper, the distance calculator is built to take x and y values directly.
Which Other Calculator Should You Use Instead?90-degree
A few related tasks are better handled by a more specific calculator.
If You Need | Use This |
A non-right triangle solved from known sides or angles | Triangle Calculator |
A square root worked out on its own | Square Root Calculator |
The slope of a line between two coordinate points | Slope Calculator |
The full set of missing sides and angles for a right triangle | Right Triangle Calculator |
Frequently Asked Questions
What Is the Pythagorean Theorem Formula?
a squared plus b squared equals c squared, where c is the hypotenuse and a and b are the two legs of a right triangle.
How Do You Find a Missing Leg Instead of the hypotenuse?
Rearrange the formula to A equals the square root of C squared minus B squared, using the known leg and the hypotenuse.
Who Invented the Pythagorean Theorem?
It is credited to the ancient Greek mathematician Pythagoras, though earlier civilizations such as the Babylonians used the same relationship centuries before him.
What Is a Pythagorean Triple?
A set of three whole numbers that satisfy a squared plus b squared equals c squared exactly, like 3, 4, and 5.
Does the Pythagorean Theorem Work for Any Triangle?
No, it only applies to right triangles, those with exactly one 90 degree angle.