Pythagorean Theorem Calculator
The Pythagorean theorem calculator needs two sides of a right triangle: leave the side you want blank. Enter all three and it checks whether the triangle is right-angled.
Source: Wolfram MathWorld – Pythagorean theorem. Updated: .
How it is calculated
The Pythagorean theorem says that in a right triangle the square of the hypotenuse equals the sum of the squares of the two legs: a² + b² = c². So c = √(a² + b²), and a leg is a = √(c² − b²). With a = 3 and b = 4, c = √25 = 5.
Legs and hypotenuse
The hypotenuse c is opposite the right angle and is always the longest side. The two sides that form the right angle are the legs a and b. Mixing them up gives wrong answers, which is why the calculator warns you if a leg is longer than c.
Formulas
- Hypotenuse: c = √(a² + b²)
- Leg: a = √(c² − b²) or b = √(c² − a²)
- Angles: α = arctan(a ÷ b), β = 90° − α
- Area: A = a × b ÷ 2 · Height to the hypotenuse: h = a × b ÷ c
- Geometric mean relations: a² = c × p, b² = c × q, h² = p × q
Exact roots and the right-angle check
When the root is not a whole number, you also get the simplified radical: with a = b = 5, c = √50 = 5√2 ≈ 7.07. Enter all three sides and the calculator applies the converse: if a² + b² = c², the triangle has a right angle. Builders use this as the 3-4-5 rule – 3 ft, 4 ft and a 5 ft diagonal, or 6-8-10 for larger layouts.
Where you need it
Rectangle or TV diagonals, how high a ladder reaches, rafter length from rise and run, and the distance between two points: √((x₂ − x₁)² + (y₂ − y₁)²). Check roots and squares with the scientific calculator, work out solids with the volume calculator, and see step-by-step radical simplification in the exponent calculator.
Frequently asked questions
How do you use the Pythagorean theorem to find c?
Square both legs, add them and take the square root: c = √(a² + b²). For a = 6 and b = 8: √(36 + 64) = √100 = 10.
How do you find a missing leg?
Square the hypotenuse, subtract the square of the known leg and take the root: a = √(c² − b²). For c = 13 and b = 12: √(169 − 144) = 5.
Does the Pythagorean theorem work for any triangle?
No, only for right triangles. For any triangle use the law of cosines, c² = a² + b² − 2ab·cos C; with C = 90° it becomes Pythagoras again.
How do I find the angles of a right triangle?
α = arctan(a ÷ b) or arcsin(a ÷ c), and β = 90° − α. In a 3-4-5 triangle: α ≈ 36.87°, β ≈ 53.13°.
How can I check that a corner is exactly 90°?
Measure all three sides and compare a² + b² with c². If they match, the angle opposite c is a right angle – that is the 3-4-5 rule used on building sites.
Sources and legal basis
- Wolfram MathWorld – Pythagorean theorem
- Wolfram MathWorld – Right triangle: angles, altitude, segments
- Wolfram MathWorld – Pythagorean triple
- Euclid, Elements I.47 and I.48 (theorem and converse; Clark University)
As of:
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