Pythagorean Theorem Calculator
Calculate the missing side of a right triangle: a² + b² = c²
Pythagorean Theorem Calculator – Find Any Side of a Right Triangle
Solve for any missing side of a right triangle using the Pythagorean theorem with our free online calculator. Enter two sides and instantly find the third with step-by-step working.
c is the hypotenuse (longest side)
Enter any two sides. Leave the side you want to find empty.
Understanding the Pythagorean Theorem
The Pythagorean theorem is one of the most fundamental relationships in geometry. It states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides. Simple to state, endlessly useful.
This calculator solves for any missing side. Enter the two sides you know, and it'll find the third – showing each step along the way. Whether you're checking homework, sizing a TV, or figuring out ladder placement, the math happens instantly.
The Formula
Where c is always the hypotenuse (longest side, opposite the 90° angle) and a and b are the legs.
Finding the Hypotenuse
Add the squares of both legs, then take the square root.
Finding Leg a
Subtract the square of the known leg from the hypotenuse squared.
Finding Leg b
Same process – subtract the known leg from the hypotenuse squared.
Pythagorean Triples
Pythagorean triples are sets of three whole numbers that satisfy a² + b² = c². These create right triangles with integer side lengths – no decimals, no rounding.
| Side a | Side b | Hypotenuse c | Verification |
|---|---|---|---|
| 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 ✓ |
| 12 | 35 | 37 | 144 + 1225 = 1369 ✓ |
Multiples of these triples also work. For example, doubling (3, 4, 5) gives (6, 8, 10), which is also a valid right triangle.
Real-World Applications
Construction and Carpentry
Builders use the 3-4-5 rule to ensure corners are perfectly square. Measure 3 feet along one wall, 4 feet along the other – if the diagonal is exactly 5 feet, the corner is a perfect 90 degrees. This works because 3² + 4² = 5².
TV and Monitor Sizing
Screens are measured diagonally. A 55-inch TV with a 16:9 aspect ratio has a width of about 47.9 inches and height of 27 inches. The diagonal (55") is the hypotenuse of that right triangle.
Ladder Placement
Need to reach a 12-foot roof? If you place the ladder base 5 feet from the wall, you need a 13-foot ladder (5² + 12² = 13²). Safety guidelines often recommend a 4:1 ratio – for every 4 feet up, move the base 1 foot out.
Navigation and GPS
The distance formula used in GPS and mapping is based on the Pythagorean theorem. To find the straight-line distance between two points, you're essentially calculating the hypotenuse of a right triangle formed by the coordinate differences.
Computer Graphics
Video games use the theorem constantly – calculating distances between objects, determining if characters are in range, collision detection, camera positioning. Every time a game checks if you're close enough to interact with something, it's using a² + b² = c².
Frequently Asked Questions
What is the Pythagorean theorem?
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c². It only works for right triangles – triangles with one 90-degree angle.
How do I identify the hypotenuse?
The hypotenuse is always the longest side and sits opposite the right angle (90°). In the formula a² + b² = c², c is always the hypotenuse. The other two sides (a and b) are called legs.
Can I use this for non-right triangles?
No. For non-right triangles, you need the Law of Cosines: c² = a² + b² - 2ab·cos(C). The Pythagorean theorem only applies when one angle is exactly 90 degrees. For oblique triangles, you'll need different formulas.
Why does the theorem work?
There are hundreds of proofs. One intuitive approach: arrange four identical right triangles to form a square. The empty space in the middle has area c². Rearrange the same triangles, and the empty space becomes two squares with areas a² and b². Same total area, so a² + b² = c².
Who discovered the Pythagorean theorem?
Although named after Greek mathematician Pythagoras (570-495 BCE), the relationship was known much earlier. Babylonian tablets from 1800 BCE show knowledge of Pythagorean triples. Pythagoras or his followers likely provided the first formal proof.
What if my answer has a decimal?
Most right triangles don't have integer sides. If you get a decimal, that's normal. For example, a triangle with legs 1 and 1 has hypotenuse √2 ≈ 1.414. This is an irrational number – it can't be written as a simple fraction.
Other Free Tools
Law Of Cosines Calculator
Law of Cosines Calculator
Law Of Sines Calculator
Law of Sines Calculator
Distance Formula Calculator
Distance Formula Calculator – Find Distance Between Two Points
Slope Calculator
Slope Calculator
Triangle Area Calculator
Triangle Area Calculator – Find Area from Base and Height
Linear Equation Solver
Linear Equation Solver – Solve ax + b = 0