Calculate the missing side of any right triangle using the Pythagorean theorem. Enter any two sides to find the third instantly. Free online tool with step-by-step solutions and visual triangle diagram.
a² + b² = c²
Please provide any 2 values below to solve the Pythagorean equation.
Enter the number before √ and the number inside √ separately.
2
(leave blank)
2√5
Example: For 2√5, enter "2" in the first box and "5" in the √ box. Leave both blank for unknown side.
So you've got a right triangle and you need to find a missing side. Maybe it's for homework, maybe you're building something, or maybe you're just curious. Either way, you're in the right place.
Our Pythagorean Theorem Calculator does all the math for you. Just plug in two sides, hit calculate, and boom - you get the third side. No algebra headaches, no late-night frustration.
But here's the thing. Just getting the answer isn't always enough. You probably want to understand how it works too, right? That's what this guide is for. We'll cover everything from the basic formula to real-world uses to common mistakes that trip people up.
Let's start simple. The Pythagorean Theorem is a rule about right triangles. You know, those triangles with one perfect 90-degree corner (like the corner of a piece of paper).
The rule says: if you square the two short sides and add them together, you get the square of the long side. In math language, it's a² + b² = c².
Here's what those letters mean:
Sounds confusing? It's actually not once you see it in action.
Using the calculator is super straightforward. Here's the step-by-step:
Make sure you're working with a right triangle. If it doesn't have a 90-degree angle, this theorem won't work. Simple as that.
Are you finding the hypotenuse (the longest side) or one of the legs? This matters because it changes how you use the formula.
Type in the two sides you know. The calculator will figure out the rest.
That's it. Your answer appears instantly. No waiting, no guessing.
Okay, let's break down a² + b² = c² in plain English.
Imagine you have a square attached to each side of your triangle. The theorem says the area of the big square (on the hypotenuse) equals the areas of the two smaller squares (on the legs) added together.
That's literally it. You're just comparing areas of squares. Once you see it that way, it makes total sense.
Here's a quick example:
Say one leg is 3 inches and the other is 4 inches. What's the hypotenuse?
3² = 9
4² = 16
9 + 16 = 25
√25 = 5
So the hypotenuse is 5 inches. See how that works?
The Pythagorean Theorem isn't just for math class. People use it every day in ways you might not expect.
Carpenters use it all the time to make sure corners are square. If you're building a deck or a fence, this theorem helps you get perfect 90-degree angles. They call it the "3-4-5 rule" because those numbers always make a right triangle.
Ever wonder how TV and phone screens are measured? It's the diagonal. So when you see a "55-inch TV," that's the hypotenuse of the screen. Manufacturers use the Pythagorean Theorem to figure that out.
Your phone's GPS uses this theorem to calculate distances between points. When it says "you're 0.3 miles from your destination," it's using a² + b² = c² under the hood.
Baseball players use it to figure out the shortest distance between bases. Soccer players use it to calculate passing angles. Even pool players use it to bank shots off the wall.
Here's where most people mess up. Don't worry, it's normal. Just watch out for these:
Remember: c is always the hypotenuse. That's the longest side, opposite the right angle. If you mix it up with a leg, your answer will be wrong.
When you add a² + b², you get c². But you need c, not c². So you have to take the square root at the end. This is probably the most common error.
The Pythagorean Theorem only works for right triangles. If your triangle doesn't have a 90-degree angle, you need a different formula (like the Law of Cosines).
If you're doing the math by hand, don't round until the very end. Rounding intermediate steps can throw off your final answer.
What if you know the hypotenuse and one leg, but need the other leg? No problem. You just rearrange the formula.
Instead of a² + b² = c², you use:
a² = c² - b² or b² = c² - a²
Here's an example:
Say the hypotenuse is 13 and one leg is 5. What's the other leg?
5² = 25
13² = 169
169 - 25 = 144
√144 = 12
So the missing leg is 12. Easy, right?
You don't need to know the proof to use the theorem. But if you're curious, here's the easiest one to understand.
Take a right triangle. Make three squares - one on each side. The square on the hypotenuse has an area of c². The squares on the legs have areas of a² and b².
Now here's the cool part. If you cut up the two smaller squares and rearrange the pieces, they fit perfectly inside the big square. That's the proof. It's called the "visual proof" or "water proof" because you can actually see it work.
There are hundreds of other proofs (even one by a US president!), but this visual one is the easiest to grasp.
Pythagoras was a Greek mathematician who lived around 500 BC. But here's the thing - he didn't actually discover this theorem. People in Babylon and China knew about it way before him.
What Pythagoras did was prove it and make it famous. He also started a weird cult-like group called the Pythagoreans who believed numbers were the basis of everything. They were obsessed with math, music, and... beans. Yeah, they had a thing about not eating beans.
So when you use this theorem, you're using math that's over 2,500 years old. Pretty wild, right?
Our calculator works great on phones. Here are a few tips:
This is just as important as knowing when to use it. The theorem only works when:
If you're dealing with angles (not sides), you need trigonometry. If your triangle isn't right, you need the Law of Cosines. And if you only know one side, you're stuck - you need more information.
Want to test yourself? Try these:
Problem 1: A right triangle has legs of 6 and 8. What's the hypotenuse?
Answer: 10
Problem 2: A right triangle has a hypotenuse of 17 and one leg of 15. What's the other leg?
Answer: 8
Problem 3: A ladder is leaning against a wall. The ladder is 10 feet long, and the bottom is 6 feet from the wall. How high up the wall does the ladder reach?
Answer: 8 feet
Problem 4: A right triangle has legs of 5 and 12. What's the hypotenuse?
Answer: 13
Problem 5: A right triangle has a hypotenuse of 25 and one leg of 7. What's the other leg?
Answer: 24
Once you've got the basics down, here are some deeper concepts:
These are sets of three whole numbers that work perfectly with the theorem. Like 3-4-5, 5-12-13, and 8-15-17. If you memorize a few, you can solve problems faster without a calculator.
This is just the Pythagorean Theorem in disguise. It finds the distance between two points on a graph. The formula is d = √[(x₂ - x₁)² + (y₂ - y₁)²]. See the a² + b² pattern?
You can use the theorem in three dimensions too. Want to find the diagonal of a box? Use the theorem twice - once for the base, then again for the full diagonal.
There are plenty of Pythagorean Theorem calculators out there. But here's what makes ours better:
Plus, you get this whole guide to help you understand what's happening. Most calculators just spit out a number and leave you confused.
Getting weird results? Here's what might be happening:
Problem: The answer is negative.
Fix: You probably entered the hypotenuse as a leg. Remember, the hypotenuse is always the longest side.
Problem: The answer is a decimal that doesn't look right.
Fix: Check your numbers. Did you accidentally swap the legs and hypotenuse?
Problem: The calculator says "invalid input."
Fix: Make sure you're only entering numbers. No letters, no symbols, no spaces.
Problem: The answer seems too big or too small.
Fix: Do a quick sanity check. The hypotenuse should be longer than either leg. If it's not, something's wrong.
The Pythagorean Theorem is one of those rare math concepts that's both simple and incredibly useful. Once you get it, you'll start seeing right triangles everywhere - in buildings, in screens, in nature.
And whenever you need to find a missing side, our calculator is here to help. No judgment, no hassle. Just the answer you need, fast.
Go ahead and try it out. Plug in some numbers and see what happens. The more you use it, the more it'll make sense.
The hypotenuse is always the longest side of a right triangle. It's also the side opposite the right angle (the 90-degree corner). If you're looking at a right triangle, the hypotenuse is the side that's diagonal, not the ones that make the square corner.
No, you can only use it on right triangles (triangles with one 90-degree angle). If your triangle doesn't have a right angle, you need a different formula called the Law of Cosines. This is a super common mistake, so double-check your triangle before you start.