Pythagorean Theorem Calculator
Find any side of a right triangle with a² + b² = c² — solve for the hypotenuse or either leg, with area, perimeter and both acute angles.
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What this tool does
Find any side of a right triangle with a² + b² = c² — solve for the hypotenuse or either leg, with area, perimeter and both acute angles. The theorem applies only to right-angled triangles. All three side fields stay visible: enter the two known sides and pick the unknown in 'Solve for' — the third field's value is ignored for the side being solved.
How to use the Pythagorean Theorem Calculator
- Enter or select solve for.
- Enter or select leg a.
- Enter or select leg b.
- Enter or select hypotenuse c.
- Read the calculated result; change any measurement to compare alternatives.
Formula
a² + b² = c². Solving for the hypotenuse: c = √(a² + b²). Solving for a leg: a = √(c² − b²) — the hypotenuse must be the longest side, so c must exceed the known leg.
- solveFor
- Solve for
- a
- Leg a
- b
- Leg b
- c
- Hypotenuse c
The theorem applies only to right-angled triangles. All three side fields stay visible: enter the two known sides and pick the unknown in 'Solve for' — the third field's value is ignored for the side being solved.
Worked example
For pythagorean theorem calculator, the following measurements illustrate the exact method: Solve for: c; Leg a: 3; Leg b: 4; Hypotenuse c: 5.
Inputs
- Solve forHypotenuse c
- Leg a3
- Leg b4
- Hypotenuse c5
Result
- Leg a3
- Leg b4
- Hypotenuse c5
- Area6
- Perimeter12
- Angle A (opposite a, degrees)36.87
- Angle B (opposite b, degrees)53.13
Results explained
- Leg a
- Leg a from the formula above. The theorem applies only to right-angled triangles. All three side fields stay visible: enter the two known sides and pick the unknown in 'Solve for' — the third field's value is ignored for the side being solved.
- Leg b
- Leg b from the formula above. The theorem applies only to right-angled triangles. All three side fields stay visible: enter the two known sides and pick the unknown in 'Solve for' — the third field's value is ignored for the side being solved.
- Hypotenuse c
- Hypotenuse c from the formula above. The theorem applies only to right-angled triangles. All three side fields stay visible: enter the two known sides and pick the unknown in 'Solve for' — the third field's value is ignored for the side being solved.
- Area
- Area from the formula above. The theorem applies only to right-angled triangles. All three side fields stay visible: enter the two known sides and pick the unknown in 'Solve for' — the third field's value is ignored for the side being solved.
- Perimeter
- Perimeter from the formula above. The theorem applies only to right-angled triangles. All three side fields stay visible: enter the two known sides and pick the unknown in 'Solve for' — the third field's value is ignored for the side being solved.
- Angle A (opposite a, degrees)
- Angle A (opposite a, degrees) from the formula above. The theorem applies only to right-angled triangles. All three side fields stay visible: enter the two known sides and pick the unknown in 'Solve for' — the third field's value is ignored for the side being solved.
- Angle B (opposite b, degrees)
- Angle B (opposite b, degrees) from the formula above. The theorem applies only to right-angled triangles. All three side fields stay visible: enter the two known sides and pick the unknown in 'Solve for' — the third field's value is ignored for the side being solved.
Frequently asked questions
In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c². The classic example is the 3-4-5 triangle: 9 + 16 = 25, so the hypotenuse is √25 = 5.
Square both legs, add them, and take the square root: c = √(a² + b²). With legs 6 and 8, c = √(36 + 64) = √100 = 10. The hypotenuse is always the side opposite the right angle and always the longest side.
Rearrange the theorem: a = √(c² − b²). The hypotenuse must be longer than the known leg — if it is not, no such right triangle exists and the calculator shows an error. For example c = 13, b = 5 gives a = √(169 − 25) = √144 = 12.
No — only right-angled triangles (one angle exactly 90°). For other triangles use the Law of Cosines, which generalizes it: c² = a² + b² − 2ab·cos C. When C = 90°, cos C = 0 and the law of cosines reduces to Pythagoras.
A set of three whole numbers satisfying a² + b² = c², such as 3-4-5, 5-12-13, 8-15-17 and 7-24-25, plus any multiple of them (6-8-10 is 3-4-5 doubled). Builders use the 3-4-5 rule to lay out perfectly square corners.
Once all three sides are known, the acute angles come from inverse trig: angle A (opposite side a) = atan(a/b). In a 3-4-5 triangle the angles are about 36.87° and 53.13°, which always sum to 90°.