Pythagorean Theorem Calculator
Solve for legs (a, b) or hypotenuse (c) in any right triangle with step-by-step arithmetic, interior angles, perimeter, and area.
Understanding the Pythagorean Theorem
Named after the ancient Greek mathematician Pythagoras, the theorem establishes a fundamental relationship among the three sides of any right triangle. It states that the square of the hypotenuse is equal to the sum of the squares of the other two sides:
For deeper analysis and related planning, you can also explore our Triangle Calculator and Area Calculator.
From this core theorem, the three variable expressions can be derived:
- To find hypotenuse:
c = √(a² + b²) - To find leg a:
a = √(c² - b²)(requires c > b) - To find leg b:
b = √(c² - a²)(requires c > a)
Trigonometric Angles and Properties
Using basic trigonometry, the acute angles in a right triangle can be directly determined from its side lengths:
- sin(α) = a / c → α = arcsin(a / c)
- sin(β) = b / c → β = arcsin(b / c)
- Because the sum of angles in any triangle is 180°, α + β = 90°.
- Area = (1/2) × a × b
Frequently Asked Questions
What is the formula for the Pythagorean theorem?
The theorem states that in a right-angled triangle, a² + b² = c², where a and b are the lengths of the two legs and c is the hypotenuse opposite the right angle.
Can you use the Pythagorean theorem for any triangle?
No. The Pythagorean theorem only applies strictly to right-angled triangles containing one 90-degree angle. For non-right triangles, the Law of Cosines is used.
What are common Pythagorean triples?
Common positive integer triples (a, b, c) include (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25).
Why must the hypotenuse c be greater than legs a and b?
The hypotenuse is opposite the largest interior angle (the 90° right angle). In Euclidean geometry, the side opposite a larger angle must always be strictly longer than sides opposite smaller angles.
What is a Pythagorean triple?
A Pythagorean triple consists of three positive integers (a, b, c) that satisfy the equation a² + b² = c². The most famous example is (3, 4, 5) because 3² (9) + 4² (16) = 5² (25).