Pythagorean Theorem Calculator: what this tool is for
The Pythagorean Theorem is one of the most celebrated fundamental theorems of Euclidean geometry. Attributed to ancient Greek mathematician Pythagoras of Samos in the 6th century BCE, it establishes that in every right-angled triangle, the square of the length of the hypotenuse (the longest side opposite the 90-degree right angle) equals the sum of the squares of the lengths of the two adjacent legs: a² + b² = c².
Geometrically, this principle proves that the area of a square constructed upon the hypotenuse equals the combined total area of squares erected upon the other two legs. This enables instant computation of any unknown triangle dimension when the other two are known: hypotenuse c = √(a² + b²) or missing leg a = √(c² - b²).
Beyond linear dimensions, right-triangle trigonometry allows you to determine acute interior angles α and β through standard sine (opposite/hypotenuse), cosine (adjacent/hypotenuse), and tangent (opposite/adjacent) ratios, alongside enclosed area A = (a × b)/2 and full perimeter P = a + b + c.
This free online Pythagorean Theorem Calculator solves for the hypotenuse, missing adjacent legs, acute degree angles, surface area, perimeter, detects integer Pythagorean triples, and renders step-by-step mathematical proofs.
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Local processing
The computation runs in your browser whenever the technology allows it.
Instant result
No waiting and no export queue: the result appears live.
Free, no account
No sign-up, no daily limit and no intrusive ads.
What this tool gives you
Omnidirectional 3-way variable solver
Solve for Hypotenuse (c = √(a² + b²)) or either adjacent leg (a or b) with exact square root step-by-step arithmetic.
Trigonometry & acute angle calculations
Automatically compute acute angles α and β (with α + β = 90°) along with sine, cosine, and tangent trigonometric ratios.
Area, perimeter & inradius / circumradius
Instantly retrieve total surface area (A = ab/2), perimeter (P = a + b + c), altitude to hypotenuse (h_c = ab/c), and incircle radius.
Pythagorean triple detection
Automatically flags whenever your inputs form an integer Pythagorean triple (such as 3-4-5, 5-12-13, 8-15-17, 7-24-25).
Real-world examples
Review standard Pythagorean triples and real-world right-triangle engineering benchmarks:
How to use Pythagorean Theorem Calculator
Select your target variable
Choose whether you are solving for the Hypotenuse (c), Side a, or Side b.
Input your two known side lengths
Enter your numerical values (e.g., a = 3 and b = 4).
Review hypotenuse, angles, and area
Inspect the calculated side length, right-triangle surface area, perimeter, and acute angles.
Inspect step-by-step arithmetic steps
View the square additions and radical extractions for homework and problem sets.
Tips and mistakes to avoid
The 3-4-5 builder's rule for perfect 90° square corners
To confirm that two walls or frames meet at a true 90-degree right angle without specialized tools, measure 60 cm along one side and 80 cm along the other: the diagonal distance between the marks must measure exactly 100 cm (1 meter)!
Finding square diagonals instantly with a√2
For any square with side length a, the diagonal cuts it into two 45°-45°-90° right triangles: c = √(a² + a²) = a√2. A 10 cm square has a diagonal of exactly 10√2 ≈ 14.14 cm.
Avoid : Applying Pythagoras to non-right triangles
The formula a² + b² = c² holds strictly when the triangle features a 90° right angle. For non-right triangles, use the Law of Cosines: c² = a² + b² - 2ab cos(γ).
Avoid : Forgetting to take the square root of the sum
Adding a² + b² calculates c², not c. You must take the square root √(a² + b²) to obtain the true length of the hypotenuse.
Frequently asked questions
Answers to the questions people ask about Pythagorean Theorem Calculator.
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Updated on August 2026. See all tools