Pythagorean Theorem Calculator
Got two sides of a right triangle and need the third? Enter the values you know and this calculator applies a² + b² = c² to find the missing side.
How a² + b² = c² Works
The Pythagorean theorem connects the three sides of any right triangle through a simple equation. The two shorter sides (legs) are labeled a and b. The longest side opposite the right angle is the hypotenuse, labeled c.
Square each leg, add the results, and you get the square of the hypotenuse. To find the actual hypotenuse length, take the square root of that sum. For example, with legs of 3 and 4: 9 + 16 = 25, and the square root of 25 is 5.
Working backward is just as easy. If you know the hypotenuse and one leg, subtract the square of the known leg from the square of the hypotenuse, then take the square root.
Where You Will Use This
Construction workers rely on 3-4-5 triangles to check if corners are square. Measure 3 feet along one wall, 4 feet along the other, and if the diagonal is exactly 5 feet, the corner is a perfect right angle.
Navigation and mapping use the theorem constantly. The straight-line distance between two points on a grid is the hypotenuse of a right triangle formed by the horizontal and vertical distances. GPS devices calculate this millions of times per day.
Home improvement projects like building decks, cutting rafters, or running diagonal bracing all start with Pythagorean calculations. Knowing the theorem saves material and prevents costly cutting errors.
Common Pythagorean Triples to Remember
Pythagorean triples are whole-number solutions that make mental math easy. The most famous is 3-4-5. Any multiple also works: 6-8-10, 9-12-15, and 30-40-50 are all valid triples.
Other useful triples include 5-12-13, 8-15-17, and 7-24-25. Carpenters and engineers memorize these to quickly verify measurements without a calculator.
While these neat whole-number sets are handy, most real-world problems produce irrational results. A triangle with legs of 1 and 1 has a hypotenuse of √2 (about 1.4142), which cannot be expressed as a simple fraction. This calculator handles both cases with precision to four decimal places.
Frequently Asked Questions
What is the Pythagorean theorem?
It states that in a right triangle, the square of the hypotenuse (longest side) equals the sum of the squares of the other two sides: a² + b² = c².
Does it work for non-right triangles?
No. The standard Pythagorean theorem only applies to triangles with a 90-degree angle. For other triangles, you need the law of cosines.
How do I find a missing leg instead of the hypotenuse?
Rearrange the formula: b = √(c² - a²). Switch to the Find Missing Side mode in this calculator to do it automatically.
What are Pythagorean triples?
These are sets of three whole numbers that satisfy a² + b² = c². Common examples include 3-4-5, 5-12-13, and 8-15-17.
Who invented the Pythagorean theorem?
While named after the Greek mathematician Pythagoras (circa 570–495 BC), evidence shows Babylonian mathematicians used this relationship over a thousand years earlier.