Pythagorean Theorem Calculator
Find the hypotenuse, solve for a missing leg, or check whether three side lengths form a right triangle using the Pythagorean theorem, with formulas, exact radicals, steps, and a live triangle diagram.
Background
For a right triangle with legs a and b and hypotenuse c, the relationship is a² + b² = c². This calculator helps students solve missing-side problems, verify side lengths, and visualize why the theorem works.
How to use this calculator
- Choose Find Hypotenuse, Find Missing Leg, or Right Triangle Check.
- Enter the known side lengths.
- Pick a unit if you want the result labeled.
- Choose Exact or a decimal output style.
- Click Calculate to see the answer, formula, live diagram, and optional steps.
- Use the quick picks for common right-triangle examples.
How this calculator works
- Find Hypotenuse: uses c = √(a² + b²).
- Find Missing Leg: uses b = √(c² − a²) after subtracting the square of the known leg from the square of the hypotenuse.
- Right Triangle Check: sorts the three side lengths, treats the largest side as c, and tests whether a² + b² = c².
- Exact mode: shows simplified radicals such as √64 = 8 or √45 = 3√5 when possible.
- Live diagram: updates the triangle drawing and highlights the side being solved for.
- Square-area aid: reinforces the relationship a² + b² = c² visually.
Formula & Equations Used
Pythagorean theorem: a² + b² = c²
Hypotenuse formula: c = √(a² + b²)
Missing leg formula: b = √(c² − a²)
Equivalent missing leg form: a = √(c² − b²)
Example Problems & Step-by-Step Solutions
Example 1 — Find the hypotenuse
If a = 3 and b = 4, find c.
- Use c = √(a² + b²).
- Substitute: c = √(3² + 4²).
- Simplify: c = √(9 + 16).
- Compute: c = √25 = 5.
Example 2 — Find a missing leg
If c = 10 and a = 6, find b.
- Use b = √(c² − a²).
- Substitute: b = √(10² − 6²).
- Simplify: b = √(100 − 36).
- Compute: b = √64 = 8.
Example 3 — Check whether a triangle is right
Check whether the side lengths 8, 15, and 17 form a right triangle.
- The largest side is 17, so test it as c.
- Compute the squares: 8² = 64, 15² = 225, and 17² = 289.
- Add the smaller squares: 64 + 225 = 289.
- Since a² + b² = c², the triangle is a right triangle.
Frequently Asked Questions
Q: What does the Pythagorean theorem say?
For a right triangle, the sum of the squares of the two legs equals the square of the hypotenuse: a² + b² = c².
Q: Which side is the hypotenuse?
The hypotenuse is the longest side of a right triangle. It is opposite the right angle.
Q: Can I use this calculator to find a missing leg?
Yes. Enter the hypotenuse and one leg, and the calculator solves for the other leg using the Pythagorean theorem.
Q: Can this calculator check whether three sides make a right triangle?
Yes. The calculator compares the sum of the squares of the two shorter sides with the square of the longest side.
Q: Does exact mode show radicals?
Yes. When possible, the calculator simplifies radicals such as √45 into 3√5.
Q: Does this calculator work for any triangle?
No. The theorem applies only to right triangles. In check mode, the calculator can test whether three side lengths form a right triangle.
All calculators