The Pythagorean Theorem Explained

How do builders make a perfect square corner with just a tape measure? They lean on the Pythagorean theorem, one of the most useful rules in all of math. For any right triangle, the theorem says a^2 + b^2 = c^2. In plain words, the two shorter sides squared and added together equal the longest side squared. Once you know this, you can find any missing side of a right triangle from the other two.

What the Pythagorean Theorem Says

The theorem links the three sides of a right triangle with one clean rule. The two shorter sides are called the legs, labeled a and b. The longest side, always opposite the right angle, is the hypotenuse, labeled c.

It is named for Pythagoras, a thinker from ancient Greece. The idea shows up in cultures far older than his, but his name stuck to it.

The rule is a^2 + b^2 = c^2. Square each leg, add those two results, and you get the hypotenuse squared. It works no matter how long or short the legs are, and it never fails for a true right triangle.

The word “squared” just means a number times itself. So 3 squared, written 3^2, is 3 x 3 = 9. Keep that in mind and the formula gets easy.

This rule is not a guess or an estimate. It holds true for every right triangle, from a tiny one on paper to a huge one across a field. That reliability is why math, science, and building all trust it.

When You Can Use It

The Pythagorean theorem works for one shape only: the right triangle. A right triangle has exactly one angle that measures 90 degrees, the square corner shown by a small box in diagrams.

If a triangle has no right angle, the formula does not apply. You would need other tools, like the law of cosines, for those shapes.

You can also flip this idea around. If three sides of a triangle happen to satisfy a^2 + b^2 = c^2, then that triangle must contain a right angle. This reverse check is called the converse of the theorem.

Here is a quick checklist before you use it:

• The triangle must have a 90 degree angle.
• The hypotenuse c is always the side opposite that right angle.
• The hypotenuse is always the longest of the three sides.
• You need to know two sides to solve for the third.

If you want to measure the straight line between two points on a grid instead, that is a close cousin. Our guide on slope and distance between two points covers that.

Find the Hypotenuse

When you know both legs, finding the hypotenuse is direct. Square each leg, add them, then take the square root of that total. The square root undoes the squaring and hands you the actual side length.

Say a right triangle has legs of 6 and 8. Plug them into the formula and solve step by step.

• Start: c^2 = a^2 + b^2
• Substitute: c^2 = 6^2 + 8^2
• Square each leg: c^2 = 36 + 64
• Add: c^2 = 100
• Take the square root: c = sqrt(100) = 10

So the hypotenuse is 10. Notice we take the positive square root, since a length cannot be negative. That is the whole method every time.

Not every answer lands on a clean whole number. If the legs were 5 and 7, then c^2 = 25 + 49 = 74, and c = sqrt(74), which is about 8.6. When that happens, round the decimal to a sensible number of places for your problem. To skip the arithmetic, a Pythagorean theorem calculator returns the exact and rounded result at once.

Find a Missing Leg

Sometimes you know the hypotenuse and one leg, and you need the other leg. You just rearrange the formula to solve for the leg you are missing.

Starting from a^2 + b^2 = c^2, subtract a^2 from both sides. That gives b^2 = c^2 – a^2, so b = sqrt(c^2 – a^2).

Say the hypotenuse c is 13 and one leg a is 5. Find the missing leg b.

• Start: b = sqrt(c^2 – a^2)
• Substitute: b = sqrt(13^2 – 5^2)
• Square each: b = sqrt(169 – 25)
• Subtract: b = sqrt(144)
• Take the square root: b = 12

The missing leg is 12. The key change is subtraction, not addition, because you are working backward from the hypotenuse.

One thing to watch: make sure the value you call c really is the hypotenuse. Since c is always the longest side, the number under the square root, c^2 – a^2, should stay positive. If it turns negative, you likely mixed up a leg with the hypotenuse.

Common Pythagorean Triples

A Pythagorean triple is a set of three whole numbers that fit the theorem perfectly. Because they use clean whole numbers, they are handy to memorize and easy to check by hand.

Here are three of the most common triples:

• 3-4-5: 3^2 + 4^2 = 9 + 16 = 25 = 5^2
• 5-12-13: 5^2 + 12^2 = 25 + 144 = 169 = 13^2
• 8-15-17: 8^2 + 15^2 = 64 + 225 = 289 = 17^2

The 3-4-5 triple answers our opening question. A builder measures 3 units along one edge and 4 units along the other. When the diagonal between those marks is exactly 5 units, the corner is a perfect 90 degrees. Carpenters, framers, and surveyors use this trick all the time.

Any multiple of a triple works too. Double 3-4-5 and you get 6-8-10, which still fits the formula.

Real Uses of the Theorem

These triples matter beyond the classroom. Screens map pixels with right triangle math, and a plane or ship can find a direct distance the same way. Ramps, roofs, and staircases all rely on the same rule to stay square and safe.

If you want to measure the space inside a shape rather than a side length, see our sibling guide on how to find the area of any shape. That is a different task from finding a side, so it uses different formulas.

Frequently Asked Questions About the Pythagorean Theorem

Sources

This article is for general education only. Always double-check the formulas and math for your own problem before you rely on the result. Reviewed for accuracy by Prof. Dr. Khalil Mudassar, PhD. Last updated September 11, 2026.