Pythagorean Theorem Calculator
Find the missing side of a right triangle. Enter two legs to get the hypotenuse, or a leg and the hypotenuse to get the other leg, with area, perimeter, and angles.
- Square the legs: 3² + 4² = 9 + 16 = 25.
- Take the square root: c = √25 = 5.
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How to Use the Pythagorean Theorem Calculator
Choose whether you are finding the hypotenuse or a leg, enter the two known side lengths, and select Calculate. The tool returns the missing side plus the triangle’s area, perimeter, and two acute angles.
- Pick “Find hypotenuse” if you know both legs, or “Find a leg” if you know one leg and the hypotenuse.
- Enter the two known positive lengths in the same units.
- Select Calculate and review the result and steps.
The theorem applies only to right triangles. When finding a leg, the hypotenuse must be the longest side, so it has to be greater than the known leg.
The Pythagorean Theorem
In a right triangle, the square of the hypotenuse c equals the sum of the squares of the two legs a and b. To find the hypotenuse, add the squares of the legs and take the square root. To find a leg, subtract the known leg’s square from the hypotenuse’s square and take the square root.
The hypotenuse is always the side opposite the right angle and is the longest side. Lengths use whatever unit the inputs share, and the area comes out in those squared units.
Pythagorean Theorem Examples
Finding the hypotenuse
For legs 3 and 4, c = √(9 + 16) = √25 = 5. This is the well-known 3-4-5 right triangle.
Finding a leg
For a leg of 6 and a hypotenuse of 10, the other leg is √(100 − 36) = √64 = 8, giving the 6-8-10 triangle.
An irrational result
For legs 1 and 1, the hypotenuse is √2 ≈ 1.41421356. Many right triangles have irrational side lengths that are best kept as radicals until the final step.
Pythagorean Triples
A Pythagorean triple is a set of three whole numbers that satisfy a² + b² = c². Common triples include 3-4-5, 5-12-13, 8-15-17, and 7-24-25. Any multiple of a triple is also a triple, so 6-8-10 and 9-12-15 both work.
Recognizing triples lets you find a missing side without a calculator. If two sides match part of a known triple, the third side often completes it.
Common Pythagorean Theorem Mistakes
Using it on a non-right triangle
The theorem holds only for right triangles. Other triangles need the law of cosines.
Adding when you should subtract
To find a leg, subtract the known leg’s square from the hypotenuse’s square. Only finding the hypotenuse uses addition.
Forgetting the square root
a² + b² gives c², not c. Take the square root at the end.
Treating the hypotenuse as a leg
The hypotenuse is the longest side, opposite the right angle. Do not place it where a leg belongs.
Mixing units
Use the same unit for both known sides so the result and area are consistent.
Pythagorean Theorem Calculator FAQs
What is the Pythagorean theorem?
In a right triangle, a² + b² = c², where c is the hypotenuse and a and b are the legs.
How do I find the hypotenuse?
Add the squares of the two legs and take the square root: c = √(a² + b²).
How do I find a missing leg?
Subtract the known leg’s square from the hypotenuse’s square and take the square root: b = √(c² − a²).
Does it work for any triangle?
No. It applies only to right triangles. Use the law of cosines for other triangles.
What is a Pythagorean triple?
Three whole numbers that satisfy the theorem, such as 3-4-5 or 5-12-13.
Are my values sent to a server?
No. The calculation runs locally in your browser.
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Reviewed by Prof. Dr. Khalil Mudassar (PhD) · Last updated: [mc_modified_date]
Sources
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Disclaimer
This calculator is intended for general educational and practical use. Review important results before relying on them in professional work.