The Pythagorean Theorem
The Pythagorean theorem states that in any right triangle, the square of the hypotenuse equals the sum of the squares of the legs:
a² + b² = c² Hypotenuse: c = √(a² + b²) Leg: a = √(c² − b²)
Identifying the parts
- Hypotenuse (c): the side opposite the right angle — always the longest side
- Legs (a, b): the two sides that form the 90° angle
Converse of the theorem
If the three sides of a triangle satisfy a² + b² = c², then the triangle is a right triangle. Use it to check whether an angle is exactly 90°.
Practical applications
- Construction: checking right angles with the (3, 4, 5) triple
- Navigation and GPS: distance between two points d = √(Δx² + Δy²)
- Physics: resultant of perpendicular forces R = √(Fx² + Fy²)
- Computer graphics: distance between pixels on the screen
Frequently asked questions
Does the Pythagorean theorem hold for any triangle?
No. Only for right triangles (those with a 90° angle). For other triangles use the Law of Cosines: c² = a² + b² − 2ab·cos(C).
How do I know which side is the hypotenuse?
The hypotenuse is always the side opposite the right angle — and always the longest of the three sides.
Can the result be irrational?
Yes! Only Pythagorean triples (special integers such as 3-4-5) give whole-number results. In most cases the hypotenuse or leg will be an irrational number such as √2 ≈ 1.414.
Did Pythagoras invent the theorem?
Not necessarily. The Babylonians and Egyptians already used this relation earlier. Pythagoras (6th century BC) is credited with systematizing it and proving it rigorously in the Western world.