📏 Pythagorean Theorem Calculator

Last updated: May 6, 2026

📏 Pythagorean Theorem Calculator

Find missing sides or verify a right triangle — with full working shown

b a c

Leave ONE field empty to find it. All values must be positive.

Enter all three side lengths. The calculator will find the longest side and test a²+b²=c².

Formula: a² + b² = c²  |  c = √(a²+b²)

The Pythagorean Theorem: What It Is, Why It Works, and How to Use It

Picture a ladder leaning against a wall. You know how tall the wall is, and you know how far the base of the ladder sits from the wall. Can you figure out how long the ladder needs to be? This is exactly the kind of problem the Pythagorean theorem was built to solve — and it has been doing so for over 2,500 years.

The Right Triangle: Where Everything Begins

Before we get into the theorem itself, let's talk about the shape it describes: the right triangle. A right triangle is simply a triangle that has one corner (one angle) that is exactly 90 degrees — a perfect square corner, like the corner of a piece of paper or the angle between a wall and the floor.

That 90-degree angle is called the right angle, and the side directly opposite to it is special. It's called the hypotenuse, and it's always the longest side of the triangle. The other two sides are called legs — often labeled a and b — and the hypotenuse is labeled c.

This labeling matters because the Pythagorean theorem is a statement about the relationship between these three measurements.

The Formula: a² + b² = c²

The theorem states something beautifully simple: if you square both legs of a right triangle and add those squares together, you get the square of the hypotenuse.

Written as a formula: a² + b² = c²

That's it. Three letters, one equation, and an enormous amount of practical power packed inside.

What does "squaring" mean? It just means multiplying a number by itself. So if leg a is 3 units long, then a² = 3 × 3 = 9. If leg b is 4 units long, then b² = 4 × 4 = 16. Add them: 9 + 16 = 25. So c² = 25, which means c = √25 = 5.

That's the famous 3-4-5 right triangle, probably the most recognized Pythagorean triple in existence. Check it: 9 + 16 = 25. Perfect.

What Is a "Pythagorean Triple"?

A Pythagorean triple is a set of three whole numbers that satisfy a² + b² = c². The 3-4-5 triangle is one. Multiply all three by 2 and you get 6-8-10 — also a right triangle. Here are a few more classics:

  • 5, 12, 13 — because 25 + 144 = 169 ✓
  • 8, 15, 17 — because 64 + 225 = 289 ✓
  • 7, 24, 25 — because 49 + 576 = 625 ✓

These triples are handy in exams because the answers come out as clean whole numbers. When you spot numbers like 5 and 12 together in a problem, the hypotenuse is almost certainly 13.

Three Things You Can Do With This Theorem

1. Find the hypotenuse when you know both legs. This is the most common use. You know a and b, and you need c. The formula gives you c = √(a² + b²).

Example: a = 6, b = 8. Then c = √(36 + 64) = √100 = 10.

2. Find a missing leg when you know one leg and the hypotenuse. Rearrange the formula: a = √(c² − b²). Essentially you subtract one square from the other.

Example: c = 13, b = 5. Then a = √(169 − 25) = √144 = 12.

3. Check whether a triangle is a right triangle. If someone gives you three side lengths and asks "is this a right triangle?", just plug the largest value in as c and check if a² + b² equals c². If it does, you have a right angle. If not, you don't.

Why Does the Formula Work? A Visual Proof

You don't need advanced math to see why a² + b² = c² is true — there's a beautiful visual argument. Imagine drawing a square on each side of a right triangle. The square built on the hypotenuse has an area equal to the combined area of the two squares built on the legs.

You can actually confirm this by cutting up the two smaller squares and rearranging those pieces to perfectly fill the big square. This "proof by rearrangement" was likely known to ancient Babylonian and Indian mathematicians long before the Greek philosopher Pythagoras popularized it in the Western world around 570–495 BCE.

Today, mathematicians have catalogued over 370 different proofs of this theorem — more than any other result in mathematics. U.S. President James Garfield even published his own original proof in 1876, making him the only U.S. president to have contributed to mathematics.

Real-World Uses You Actually Encounter

The theorem shows up constantly in everyday situations, often without people realizing it:

  • Construction: Builders use the 3-4-5 rule to check that corners are perfectly square before laying foundations or tiling floors.
  • Screen sizes: When manufacturers say a TV is "55 inches," they mean the diagonal of the screen — which is the hypotenuse of a rectangle.
  • GPS navigation: Calculating straight-line distance between two map coordinates uses a version of this same formula.
  • Video games: Game engines calculate distances between characters and objects using the Pythagorean theorem dozens of times per second.
  • Ramps and roofs: Architects compute slope lengths, rafter lengths, and ramp distances using right-triangle geometry constantly.

Common Mistakes Students Make

The most frequent error is forgetting which side is the hypotenuse. The hypotenuse is always opposite the right angle and always the longest side. If you accidentally assign the longest side as one of the legs in your formula, you'll get a wrong answer every time.

Another pitfall: trying to use the theorem on non-right triangles. The formula a² + b² = c² only holds for right triangles. An equilateral triangle, a scalene triangle without a 90° angle — none of those follow this rule.

Finally, students sometimes forget to take the square root at the end. Once you calculate c², you still need to find c itself by taking √c². Forgetting this step leaves you with the square of the answer, not the answer.

Quick Exam Tips

Memorize these three triples before any geometry exam: 3-4-5, 5-12-13, and 8-15-17. Any multiple of these (like 6-8-10 or 10-24-26) also works. When you see two sides matching a triple, write down the third immediately — no calculation needed.

Also remember: √2 ≈ 1.414. Why? Because an isosceles right triangle (both legs equal 1) has hypotenuse √(1² + 1²) = √2. This ratio appears constantly in geometry, especially with 45-45-90 triangles.

The Pythagorean theorem is one of those rare mathematical facts that genuinely shows up everywhere — from your phone to the roof above your head. Once you understand it, you start seeing right triangles hiding in the world around you constantly.

FAQ

Which side is the hypotenuse in the Pythagorean theorem?
The hypotenuse is always the side directly opposite the right angle (the 90° corner). It is always the longest of the three sides and is labeled 'c' in the formula a² + b² = c². If you're not sure which side to call c, just pick the longest one — that's always the hypotenuse in a right triangle.
Can I use the Pythagorean theorem on any triangle?
No — the formula a² + b² = c² only works for right triangles (triangles with exactly one 90-degree angle). For other triangles (acute or obtuse), you need different rules like the Law of Cosines. However, you can use the theorem in reverse to check whether a triangle is a right triangle: if a² + b² equals c² for the three sides, then it must have a right angle.
What is a Pythagorean triple and why does it matter for exams?
A Pythagorean triple is a set of three positive whole numbers that satisfy a² + b² = c², like 3-4-5, 5-12-13, or 8-15-17. They matter for exams because the answer comes out as a neat whole number — no messy square roots. When you spot 3 and 4 as two sides, the third is immediately 5. Memorizing the common triples (and their multiples like 6-8-10) can save you significant calculation time during tests.
How do I find a missing leg (not the hypotenuse) using this theorem?
Rearrange the formula. If you know hypotenuse c and leg b, then a = √(c² − b²). Essentially: square the hypotenuse, square the known leg, subtract the smaller from the larger, then take the square root. For example, if c = 10 and b = 6, then a = √(100 − 36) = √64 = 8. Always make sure c is the hypotenuse (the largest side) before subtracting.
Why does the calculator show slightly rounded decimal answers sometimes?
Most right triangles do not have clean whole-number answers — for example, if a = 1 and b = 1, then c = √2 ≈ 1.414213... This is an irrational number that never ends or repeats. The calculator rounds to 6 decimal places for display. In school work, you can leave answers as exact square roots (like √2) unless the problem asks for a decimal approximation.
What is the real-world use of the Pythagorean theorem outside school?
Builders use it to verify that corners are perfectly square (the 3-4-5 method). Architects compute roof rafter lengths and ramp distances. TV screen sizes advertised in inches refer to the diagonal — the hypotenuse of the rectangular screen. Navigation apps calculate straight-line distances between coordinates using this formula. Even video games rely on it to compute distances between objects many times per second.