How the Pythagorean Theorem Works, with Proofs and Examples

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. If the legs have lengths a and b, and the hypotenuse has length c, then a² + b² = c². It is one of the most widely used results in mathematics, and it links the geometry of shapes to the arithmetic of numbers. This guide explains why it holds, shows a proof you can follow with paper, and gives examples you can check yourself.

What the theorem says

A right triangle has one angle of exactly 90 degrees. The two sides that meet at that angle are called legs. The side opposite the right angle is the longest side and is called the hypotenuse.

The theorem applies only to right triangles. For a triangle with sides 4, 5, and 6, the squares do not satisfy the equation, so the triangle is not right angled. Checking the relationship is a quick way to test whether a triangle has a right angle.

A proof by areas

Many proofs exist, and more than a hundred are recorded. A short and visual one uses areas. Take a large square with side length a + b. Place four copies of the right triangle, each with legs a and b, in the corners so that they leave a tilted square in the middle.

The large square has area (a + b)². It can also be seen as the four triangles plus the tilted square. Each triangle has area ab/2, so the four triangles together have area 2ab. The tilted square has side c, because each of its sides is the hypotenuse, so its area is c².

Setting the two expressions equal gives (a + b)² = 2ab + c². Expanding the left side yields a² + 2ab + b² = 2ab + c². Subtracting 2ab from both sides leaves a² + b² = c². The proof works because the arrangement of the same pieces produces the same total area in two different ways.

Why the proof works

The key idea is that area is preserved when shapes are rearranged. The two ways of viewing the big square are two accountings of the same region. Any equation that comes out of comparing them must therefore be true, provided the geometry of the arrangement is correct.

This style of reasoning, comparing two decompositions of one region, appears throughout geometry. It gives a visual justification that helps learners remember the result rather than memorize it as an isolated rule.

Worked examples

Suppose a right triangle has legs 3 and 4. Then c² = 3² + 4² = 9 + 16 = 25, so c = 5. This is the most famous small triangle.

Suppose instead the hypotenuse is 13 and one leg is 5. Then the other leg satisfies b² = 13² - 5² = 169 - 25 = 144, so b = 12.

Suppose a ladder 10 metres long leans against a wall with its foot 6 metres from the base. The height reached on the wall is h² = 10² - 6² = 100 - 36 = 64, so h = 8 metres. Problems like this show up in construction, navigation, and surveying.

Common Pythagorean triples

A Pythagorean triple is a set of three positive whole numbers that satisfy a² + b² = c². The table below lists the most useful ones. Multiplying any triple by a whole number gives another triple, so (6, 8, 10) is simply twice (3, 4, 5).

Legs a and b	Hypotenuse c	Check
3 and 4	5	9 + 16 = 25
5 and 12	13	25 + 144 = 169
8 and 15	17	64 + 225 = 289
7 and 24	25	49 + 576 = 625
20 and 21	29	400 + 441 = 841

Triples are useful for building right angles on site without a protractor. Marking 3 units, 4 units, and 5 units along two lines forms an exact right angle. Carpenters have used this method for centuries.

Distance in the coordinate plane

The theorem also gives the distance between two points on a grid. If one point is at (x1, y1) and another is at (x2, y2), the horizontal difference is the leg a and the vertical difference is the leg b. The straight-line distance is the hypotenuse, so d = √((x2 - x1)² + (y2 - y1)²).

For points (1, 2) and (4, 6), the differences are 3 and 4, so the distance is 5. This formula underlies much of analytic geometry and the way computers measure separation between objects in graphics and mapping software.

Converse and extensions

The converse is also true. If a triangle satisfies a² + b² = c², where c is the longest side, then the triangle is right angled. This property lets you test whether an angle is exactly 90 degrees using only side lengths.

The theorem extends to three dimensions. The distance across a rectangular box with sides x, y, and z is √(x² + y² + z²). A diagonal of a cube with side s is therefore s√3, which follows from applying the theorem twice.

Common mistakes

The most frequent error is applying the theorem to a triangle that is not right angled. Confirm the right angle first, either from the problem statement or from a diagram with a square corner marked.

A second error is forgetting which side is the hypotenuse. The hypotenuse must be the side opposite the right angle, and it is always the longest side. Substituting a leg as c can leave a negative number under the square root when solving for the other leg, which signals the mistake.

A third error is adding the legs instead of their squares. The sum a + b does not equal c, so always square each term before adding.

Key takeaways

The Pythagorean theorem states that a² + b² = c² for every right triangle, where c is the hypotenuse. Its area proof explains why the relationship holds, and that picture is easier to remember than the formula alone.

Use the theorem to find a missing side, to measure distance on a coordinate grid, and to test whether an angle is a right angle. The converse makes this test reliable.

Learn a few common triples such as 3, 4, 5 and 5, 12, 13. They save time and let you check results at a glance.