Pythagorean Theorem Calculator
Calculate hypotenuse or missing leg using the Pythagorean theorem a²+b²=c². Find triangle area and perimeter at the same time.
Result
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Triangle Area —
Perimeter —
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Hypotenuse c
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Triangle Area —
Perimeter —
Pythagorean Triple? —
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Core Results
Hypotenuse c —
Area —
Perimeter —
Angles
Angle A (at leg a) ° —
Angle B (at leg b) ° —
Special Properties
Inradius —
Circumradius —
Pythagorean Triple? —
Altitude to Hypotenuse —
How to Use This Calculator
Select Solve For (hypotenuse or a missing leg), enter the two known values, and get the result plus the triangle's area and perimeter.
Formula
Hypotenuse: c = √(a²+b²) • Leg: a = √(c²−b²)
Example
a=3, b=4 → c=5, Area=6, Perimeter=12
Frequently Asked Questions
- The Pythagorean theorem states that in any right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c², where c is the hypotenuse (the side opposite the right angle). This theorem applies only to right triangles — it will give incorrect results if used on acute or obtuse triangles. It is one of the most widely used formulas in mathematics, underlying distance calculations, navigation, construction, and trigonometry.
- To find the hypotenuse c when both legs a and b are known, use c = √(a² + b²). Example: a = 3, b = 4 → c = √(9 + 16) = √25 = 5. Another example: a = 5, b = 12 → c = √(25 + 144) = √169 = 13. The hypotenuse is always the longest side of a right triangle. A common error is using the formula for non-right triangles — always verify you have a right angle before applying this formula.
- To find a missing leg a when the hypotenuse c and the other leg b are known: a = √(c² − b²). Example: hypotenuse = 13, one leg = 5 → a = √(169 − 25) = √144 = 12. Another example: hypotenuse = 10, one leg = 6 → a = √(100 − 36) = √64 = 8. Notice that c must be larger than b for the result to be real. If you enter a leg larger than the hypotenuse, the calculation gives the square root of a negative number — which is undefined in real arithmetic.
- A Pythagorean triple is a set of three positive integers (a, b, c) that exactly satisfy a² + b² = c², forming a perfect right triangle with no rounding. Common triples: (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25), (9, 40, 41), and (20, 21, 29). All multiples of a triple are also triples: (6, 8, 10), (9, 12, 15), etc. Pythagorean triples are useful in construction and carpentry for ensuring perfectly square corners — if you measure 3 ft, 4 ft, and the diagonal is exactly 5 ft, the corner is a perfect right angle.
- No — the Pythagorean theorem (a² + b² = c²) applies strictly to right triangles. For acute triangles, a² + b² > c² (both legs squared sum to more than the hypotenuse squared). For obtuse triangles, a² + b² < c². If you need to solve an arbitrary triangle given three sides, use the Triangle Calculator which applies Heron's formula (for area) and the Law of Cosines (for angles). The Pythagorean theorem is a special case of the Law of Cosines when the included angle is 90°.