Enter any two sides of a right triangle and this calculator finds the missing side instantly. It uses the classic formula a² + b² = c² to compute the hypotenuse or either leg, with step-by-step solutions so you can follow the math.
A right triangle is a triangle with one 90 degree angle. The two shorter sides (called legs) meet at the right angle. The longest side, opposite the right angle, is the hypotenuse.
This free pythagorean theorem calculator handles all three cases: find the hypotenuse from two legs, or solve for a missing leg when you know the hypotenuse and one leg.
How to Use the Pythagorean Theorem to Find the Missing Side
- Identify which sides you know. Label the two legs as a and b, and the hypotenuse as c.
- Enter the two known side lengths into the calculator.
- Click calculate. The tool returns the missing side, the calculation steps, and the triangle's perimeter.
Which side is the hypotenuse? It is always the side directly across from the right angle. It is also always the longest side of a right triangle. If you are unsure, look for the side that does not touch the 90 degree angle.
The calculator works whether you need to find the hypotenuse or solve for a missing leg. Just leave the unknown field empty and fill in the two sides you have.
The Pythagorean Theorem Formula: a² + b² = c²
The Pythagorean theorem states that in any right triangle, the square of the hypotenuse equals the sum of the squares of the two shorter sides.
Written as an equation:
a² + b² = c²
- a and b are the legs of the right triangle (the perpendicular sides).
- c is the hypotenuse (the longest side, opposite the right angle).
The formula works in one direction to find the hypotenuse and in reverse to find either leg. This relationship is the foundation of right triangle geometry and dates back to the ancient Greek mathematician Pythagoras.
Is the Pythagorean theorem only for right triangles? Yes. The a² + b² = c² equation applies only when one angle is exactly 90 degrees. For other triangles, you would need the law of cosines, which is a different formula entirely.
Find the Hypotenuse of a Right Triangle
To find the hypotenuse, square each leg, add the results, and take the square root.
c = √(a² + b²)
Example: legs are 7 and 9.
- 7² = 49
- 9² = 81
- 49 + 81 = 130
- c = √130 ≈ 11.40
The hypotenuse is approximately 11.40 units.
Use the Pythagorean Theorem When You Know Two Sides
When you know both legs, the calculation is straightforward: plug them in as a and b, then solve for c. The pythagorean theorem calculator to find the hypotenuse does exactly this in a single step.
What if you know the hypotenuse and one leg? Then you are solving for the other leg, which is covered in the next section.
Solve for a Side of a Right Triangle Using the Equation
Rearrange the pythagorean equation to isolate the unknown leg:
a = √(c² − b²)
Example: hypotenuse is 13, one leg is 5.
- 13² = 169
- 5² = 25
- 169 − 25 = 144
- a = √144 = 12
The missing leg is 12 units. This is actually one of the classic Pythagorean triples (5, 12, 13).
What about an isosceles right triangle with hypotenuse 10? Both legs are equal, so a = b. The formula becomes 2a² = 100, giving a = √50 ≈ 7.07.
Step-by-Step Solutions for the Right Triangle Calculator
This calculator shows every intermediate step:
- Identify knowns. The tool recognizes which two values you entered.
- Square each known side. It displays a² and b² (or c² and the known leg²).
- Add or subtract. It shows the sum of the squares when finding the hypotenuse, or the difference when finding a leg.
- Take the square root. The final result is the length of the missing side, rounded to a reasonable number of decimal places.
Step-by-step solutions help you verify your own homework or confirm a measurement before cutting material for a project.
Verify a Triangle Using the Pythagorean Theorem
Have three side lengths and want to check whether they form a right triangle? Plug all three into the calculator and it will verify.
The test is simple:
- Square the two shorter sides and add them together.
- Square the longest side.
- If the sum of the squares of the two shorter sides equals the square of the longest side, the triangle is a right triangle.
Example: sides 6, 8, 10.
- 6² + 8² = 36 + 64 = 100
- 10² = 100
- They match. This is a right triangle.
If the numbers do not match, the three lengths form a different type of triangle (or no triangle at all if they violate the triangle inequality).
Pythagorean Triples
Pythagorean triples are sets of three whole numbers that satisfy a² + b² = c² exactly, with no decimals or rounding.
Common examples:
- 3, 4, 5
- 5, 12, 13
- 8, 15, 17
- 7, 24, 25
Any multiple of a triple also works. For instance, 3, 4, 5 scaled by 2 gives 6, 8, 10, which is also a valid triple.
Pythagorean triples are useful in construction, carpentry, and coordinate geometry because they produce right angles with clean whole number measurements. The 3, 4, 5 triple is the quickest way to confirm a square corner on a job site.
Frequently Asked Questions
How Do I Use the Pythagorean Theorem Calculator to Find the Hypotenuse?
Enter the two leg lengths into the a and b fields and leave the hypotenuse field empty. Click calculate. The tool squares both legs, adds the results, and returns the square root as the hypotenuse length. You also get the full step-by-step solution.
Can This Calculator Solve for Any Missing Side of a Right Triangle?
Yes. It solves for any one missing side when you provide the other two. That includes finding a missing leg when you know the hypotenuse and one leg. Enter your two known values, leave the third blank, and the calculator does the rest.
Does this calculator work for any triangle? No. The a² + b² = c² formula applies only to right triangles (those with a 90 degree angle). For non-right triangles, the law of cosines is the correct approach.
What Is the Pythagorean Theorem Formula for a² + b² = c²?
The formula means: in a right triangle, the sum of the squares of the two legs equals the square of the hypotenuse. The variable a and b represent the two perpendicular sides, and c is the longest side opposite the right angle. Rearranging the same equation lets you solve for any one of the three sides.
How Do I Verify a Right Triangle with This Calculator?
Enter all three side lengths. The calculator squares the two shorter sides, adds them, and compares that sum to the square of the longest side. If they are equal, the three lengths form a right triangle. If not, the triangle is not a right triangle. This is a quick way to check whether three side lengths form a right triangle before starting further calculations.