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Euclidean Geometry
Pythagorean Theorem Calculator (a² + b² = c²)
Solve right triangle side lengths, angles, perimeter, and area instantly. Calculate the hypotenuse from two legs, find missing adjacent/opposite sides, verify Pythagorean triples, and review geometric derivations.
Calculated Hypotenuse (c)
c = 5.00
✓ Perfect Pythagorean Triple (3, 4, 5)
Side a
3.00
Side b
4.00
Hypotenuse c
5.00
Triangle Area
6.00
Perimeter
12.00
Angle α / β
36.87° / 53.13°
Step-by-Step Calculation:
The Pythagorean Theorem: Geometric Proofs & Trigonometry
Named after the ancient Greek philosopher Pythagoras of Samos (~570–495 BC), the Pythagorean theorem states that in any right-angled triangle in a flat Euclidean plane, the square of the length of the hypotenuse (the side opposite the 90-degree right angle) is strictly equal to the sum of the squares of the lengths of the remaining two legs.
The Fundamental Theorem
a² + b² = c²
Famous Integer Pythagorean Triples
A Pythagorean triple consists of three positive integers (a, b, c) that satisfy the theorem perfectly with zero fractional decimals. Commonly encountered triples in geometry and carpentry:
- (3, 4, 5): The most famous triple (9 + 16 = 25). Multiples like (6, 8, 10) and (9, 12, 15) are also valid right triangles. Builders use the 3-4-5 rule to square foundation corners.
- (5, 12, 13): 25 + 144 = 169.
- (8, 15, 17): 64 + 225 = 289.
- (7, 24, 25): 49 + 576 = 625.
Frequently Asked Questions
No. The classical theorem only applies to right triangles with a 90° angle. For oblique triangles (acute or obtuse), mathematicians use the Law of Cosines: c² = a² + b² - 2ab×cos(C).
No. The hypotenuse is always strictly the longest side of a right triangle, because it opposes the largest interior angle (90 degrees).
Educational Disclaimer: Formulas compute Euclidean distance in 2D space.