The Pythagorean theorem: what it says and how to use it

What the theorem says

A right triangle has one 90° angle. The two sides that form that angle are the legs, usually called a and b. The side across from the right angle is the hypotenuse, c. It is always the longest side.

The theorem says a² + b² = c². In picture form: build a square on each side of the triangle. The areas of the two squares on the legs add up exactly to the area of the square on the hypotenuse.

It works only for right triangles. For any other triangle, a² + b² and c² are not equal, and that difference is useful too, as the converse section shows.

Why it is true: a proof you can draw

There are hundreds of proofs. One of the clearest uses four copies of the same right triangle, with legs a and b and hypotenuse c.

Draw a large square with side a + b. Place the four triangles inside it, one in each corner, so that their hypotenuses form a tilted square in the middle. That middle square has side c, so its area is c².

Now count the area of the big square two ways. Its side is a + b, so its area is (a + b)², which expands to a² + 2ab + b². It is also made of the middle square plus four triangles, each with area ab/2, so its area is c² + 2ab. Setting the two equal and taking 2ab from both sides leaves a² + b² = c².

Rearranging the same four triangles in the big square leaves two uncovered squares, of areas a² and b², instead of one of area c². The uncovered area did not change, which is the same fact seen without algebra.

Worked example: finding a missing leg

When the hypotenuse is known, subtract: the unknown leg squared is c² minus the known leg squared. Adding here is the most common error, and it would give a leg longer than the hypotenuse, which is impossible.

Pythagorean triples

Some right triangles have whole number sides. These sets are called Pythagorean triples, and recognising them saves time on tests:

• 3, 4, 5, and its multiples such as 6, 8, 10 and 9, 12, 15.
• 5, 12, 13.
• 8, 15, 17.
• 7, 24, 25.
• 20, 21, 29.

The converse: is it a right triangle?

The converse of the theorem is also true: if the sides of a triangle satisfy a² + b² = c², the triangle has a right angle. To test three lengths, let c be the longest, then compare a² + b² with c²:

• Equal: right triangle. For 7, 24 and 25: 49 + 576 = 625 and 25² = 625, so it is right.
• a² + b² greater than c²: every angle is acute. For 6, 7 and 8: 36 + 49 = 85, which is more than 64.
• a² + b² less than c²: the triangle has an obtuse angle. For 4, 5 and 8: 16 + 25 = 41, which is less than 64.

Worked example: the distance between two points

Any two points on a grid are the ends of the hypotenuse of a right triangle whose legs run along the grid lines. The distance formula, d = √((x₂ − x₁)² + (y₂ − y₁)²), is the Pythagorean theorem written in coordinates.

Worked example: the diagonal of a box

In three dimensions, use the theorem twice: once across the floor of the box, then once up to the opposite top corner. The result is d² = l² + w² + h².

Common mistakes and how to check your work

Most errors come from a few habits:

• Adding when you should subtract. If the hypotenuse is given, the leg is √(c² − a²).
• Labelling the wrong side as c. The hypotenuse is opposite the right angle and is always the longest side.
• Taking a square root term by term: √(9 + 16) is √25 = 5, not 3 + 4 = 7.
• Using the theorem on a triangle with no right angle.
• Forgetting to take the square root at the end, and giving c² as the answer.
• Check every answer: the hypotenuse must be longer than each leg but shorter than the two legs added together, and the squares must add up when you substitute back in.