The Pythagorean Theorem: Formulas, Proofs, and Uses
The Pythagorean theorem is a fundamental relation in Euclidean geometry among the three sides of a right triangle. Originally attributed to the Greek philosopher Pythagoras (though known to ancient Babylonians and Indians much earlier), it is one of the most widely used mathematical formulas in construction, engineering, and navigation.
The Mathematical Formula
The theorem states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
Formula: a² + b² = c²
How to Find the Hypotenuse (c)
If you know the lengths of the two shorter sides (a and b), square both of them, add the results together, and then find the square root of that sum.
Example: If a = 3 and b = 4.
3² + 4² = 9 + 16 = 25. The square root of 25 is 5. Therefore, c = 5.
How to Find a Missing Leg (a or b)
If you already know the hypotenuse (c) and one leg (a), rearrange the formula to solve for the missing leg: b = √(c² - a²). Simply square the hypotenuse, subtract the square of the known leg, and take the square root of the result.