Pythagorean Theorem Calculator
Find the missing side of any right triangle using the Pythagorean theorem (a² + b² = c²). Enter any two sides and this calculator solves for the third, along with the triangle's area and perimeter.
The Pythagorean theorem is one of the most fundamental and widely-used theorems in all of mathematics. For any right triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides.
The Formula: a² + b² = c²
Where a and b are the legs (the sides forming the right angle) and c is the hypotenuse (always the longest side).
Solving for Each Side:
- Hypotenuse: c = sqrt(a² + b²). Example: If a = 3 and b = 4, then c = sqrt(9 + 16) = sqrt(25) = 5.
- Leg a: a = sqrt(c² - b²). Example: If c = 13 and b = 5, then a = sqrt(169 - 25) = sqrt(144) = 12.
- Leg b: b = sqrt(c² - a²). Same process as above, solving for the other leg.
Pythagorean Triples:
Some right triangles have all integer side lengths. These are called Pythagorean triples. Common examples include (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25), and (20, 21, 29). Any multiple of a Pythagorean triple is also a triple: (6, 8, 10) is a multiple of (3, 4, 5).
Applications:
The Pythagorean theorem is used in construction (ensuring right angles), navigation (distance calculations), computer graphics (pixel distances), physics (vector magnitudes), and countless other fields.