Pythagorean theorem calculator
The missing side of a right triangle, with a² + b² = c² worked out
Type any two sides and leave the one to find empty. The example is a 3, 4, 5 triangle.
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- The two other angles
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- Area
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- Perimeter
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Working, step by step
How the missing side is found
In a right triangle (one with a 90° corner), the two shorter sides a and b and the longest side c, called the hypotenuse, always satisfy a² + b² = c². The hypotenuse is the side opposite the right angle.
Finding the hypotenuse
Square the two shorter sides, add them and take the square root. For sides 3 and 4: 9 + 16 = 25, and √25 = 5.
Finding a shorter side
Square the hypotenuse, take away the square of the side you know, and take the square root. With c = 13 and a = 5: 169 − 25 = 144, and √144 = 12. The hypotenuse must be the longest side, so the page refuses a "hypotenuse" that is shorter than the other side.
Exact and rounded answers
When the result is a whole number or a fraction, it is shown exactly. Otherwise the square root is simplified where it can be: sides 1 and 1 give a hypotenuse of √2, about 1.414213562; sides 2 and 2 give √8 = 2√2. Decimals are rounded to 10 significant digits.
The other results
The two angles that are not the right angle add up to 90°. Each comes from the inverse tangent of opposite ÷ adjacent. The area is a × b ÷ 2 and the perimeter is a + b + c.
Units
Use any unit, such as inches, feet, centimeters or meters, but use the same one for both sides you type. The answer is in that unit, and the area is in square units. Builders use the 3, 4, 5 rule to check that a corner is square.
Formula and sources
- Pythagorean theorem: Common Core 8.G.B.7 (unknown side lengths in right triangles).
- Angles from the inverse tangent: Common Core HSG-SRT.C.8.
- Checked in test_maths.js: 3, 4 → 5; 5, c = 13 → 12; 1, 1 → √2 ≈ 1.414213562; 2, 2 → 2√2; 8, 15 → 17; a hypotenuse shorter than a side is refused.