Pythagorean Theorem Calculator
Find any side of a right triangle using a² + b² = c².
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The Pythagorean Theorem
In 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: a² + b² = c². This theorem is named after the ancient Greek mathematician Pythagoras (c. 570–495 BC), though the relationship was known to Babylonian and Indian mathematicians earlier.
Finding the Hypotenuse
c = √(a² + b²). For the classic 3-4-5 right triangle: c = √(9+16) = √25 = 5. Common Pythagorean triples (integer solutions): (3,4,5), (5,12,13), (8,15,17), (7,24,25), (6,8,10).
Finding a Leg
a = √(c² − b²). If hypotenuse = 13 and one leg = 5: a = √(169−25) = √144 = 12.
Pythagorean Triples
A Pythagorean triple is three positive integers (a, b, c) satisfying a²+b²=c². The formula to generate them: for any m > n > 0, a=m²−n², b=2mn, c=m²+n². For m=2, n=1: a=3, b=4, c=5.
Applications
Pythagorean theorem is used in construction (checking right angles, staircase calculations), navigation (straight-line distances), coordinate geometry (distance formula = √((x₂−x₁)²+(y₂−y₁)²)), physics, and 3D geometry.
Worked Example: Ladder Against a Wall
A ladder leans against a wall. The base of the ladder is 6 feet from the wall, and the ladder reaches 8 feet up the wall. How long is the ladder?
The wall, ground, and ladder form a right triangle with legs a = 6 and b = 8. Using the Pythagorean theorem: c = √(6² + 8²) = √(36 + 64) = √100 = 10 feet. This is the classic 3-4-5 triple scaled by 2: (6, 8, 10).
Common Pythagorean Triples
| Leg a | Leg b | Hypotenuse c | Verify |
|---|---|---|---|
| 3 | 4 | 5 | 9 + 16 = 25 ✓ |
| 5 | 12 | 13 | 25 + 144 = 169 ✓ |
| 8 | 15 | 17 | 64 + 225 = 289 ✓ |
| 7 | 24 | 25 | 49 + 576 = 625 ✓ |
| 9 | 40 | 41 | 81 + 1600 = 1681 ✓ |
Frequently Asked Questions
a² + b² = c². In a right triangle, the sum of the squares of the two legs equals the square of the hypotenuse.
(3,4,5), (5,12,13), (8,15,17), (7,24,25), (6,8,10). Any multiple of these also works: (6,8,10), (9,12,15).
c = √(a² + b²). For legs 5 and 12: c = √(25+144) = √169 = 13.
Construction (squaring corners: if the diagonal = √(side1²+side2²), the corner is 90°), navigation, physics, and computer graphics all use it constantly.
Yes. The space diagonal of a box with dimensions a×b×c is d = √(a²+b²+c²). Just extend the theorem one dimension.
Square all three sides and check if the sum of the two smaller squares equals the largest square. For sides 9, 40, 41: 9² + 40² = 81 + 1600 = 1681 = 41². Yes, it is a right triangle.
The distance formula between two coordinate points (x₁,y₁) and (x₂,y₂) is d = √((x₂−x₁)² + (y₂−y₁)²). It is directly derived from the Pythagorean theorem where the horizontal difference and vertical difference form the two legs of a right triangle.