The Pythagorean theorem

A right triangle is a triangle with one right angle, a corner of exactly 90°, like the corner of a book. Its three sides have a special connection that people have known for more than 2,500 years. It is named after the Greek thinker Pythagoras, but it was already used in Babylon long before him, and it was also known in ancient China. With it, you can find any side of a right triangle when you know the other two.

The names of the sides

The two sides that meet at the right angle are called the legs. We call them a and b. The longest side is across from the right angle. It is called the hypotenuse, and we call it c. The hypotenuse is always the longest side, so your answer for c must always be bigger than both legs.

The rule

In words: square the first leg, square the second leg, and add them. The total is the square of the hypotenuse. Remember that squaring a number means multiplying it by itself, so 5² = 5 × 5 = 255² = 5 × 5 = 25.

Why is this true? Draw a square on each side of the triangle. The area of the square on the hypotenuse is exactly the same as the areas of the two smaller squares added together. For a triangle with legs 3 and 4, the small squares have areas 9 and 16. Together that is 25, and the square on the hypotenuse has area 25 too, so the hypotenuse is 5.

Finding the hypotenuse

When you know both legs, follow these steps:

1. Square each leg.
2. Add the two squares.
3. Take the square root of the total. That is c.

Finding a leg

When you know the hypotenuse and one leg, you subtract instead of adding. The square of the hypotenuse is the big total, and you take away the part you already know.

Pythagorean triples

Some right triangles have three whole-number sides. These sets of numbers are called Pythagorean triples. If you know the most common ones, you can often answer without any working at all.

Diagonals of rectangles and squares

A diagonal cuts a rectangle into two right triangles. The sides of the rectangle are the legs, and the diagonal is the hypotenuse. So the diagonal of a rectangle that is 4 by 3 is √(16 + 9) = √25 = 5√16 + 9 = √25 = 5.

Does the answer make sense?

• The hypotenuse must be longer than each leg.
• The hypotenuse must be shorter than the two legs added together. With legs 6 and 8, the answer must be between 8 and 14. It is 10.
• If you get a number under a square root that is not a square number (like 49 or 100), for example √50√50, the answer is not a whole number. That is fine, but check your squares once more.