What Is the Pythagorean Theorem?

What each side is doing

The side opposite the right angle has a special name: the hypotenuse. It is always the longest side. The other two sides are called the legs because they meet to make the right angle. Before writing any numbers, find the square corner and then look straight across from it. That simple check tells you which length belongs in the c position.

Suppose a right triangle has legs of 3 units and 4 units. The equation becomes 3² + 4² = c², so 9 + 16 = c² and 25 = c². Since 5 × 5 is 25, the missing side is 5 units. The calculation makes sense because the 5-unit side is opposite the right angle and is longer than either leg.

You can also use the rule when the hypotenuse is known. If c is 13 and one leg is 5, write 5² + b² = 13². That gives 25 + b² = 169, so b² = 144 and b = 12. Notice the order: square first, isolate the unknown square, and only then take the square root.

The theorem also explains distance on a coordinate grid. A horizontal change and a vertical change make the two legs of a hidden right triangle. The straight segment between the points is its hypotenuse. That is why the distance formula contains squared changes: it is the Pythagorean Theorem wearing coordinate-geometry clothing.

A good sketch does not need to look perfect. Mark the right angle, label the three sides, and write the relationship beside the picture. If three given lengths do not satisfy the equation, do not force the result. Either the triangle is not right, a label was copied incorrectly, or one of the measurements has been used in the wrong place.

• In a right triangle, a squared plus b squared equals c squared.
• The theorem applies only to right triangles.
• The hypotenuse is the side opposite the right angle and is the side whose square stands alone in the equation.
• The converse allows side lengths to prove that a triangle is right.

Where this relationship helps

• Use the Pythagorean Theorem to find a missing side in a right triangle.
• Use it in coordinate geometry to derive distance between points.
• Use it in real-world measurement problems involving perpendicular directions.

Errors that change the answer

• Do not apply the theorem to a triangle unless a right angle is present or proven.
• Do not put the wrong side in the hypotenuse position of the equation.
• Do not forget to square the side lengths before adding or comparing them.