Pythagorean Theorem

What is Pythagorean Theorem

The Pythagorean theorem, or Pythagoras' theorem, is a fundamental theorem of mathematics that describes the relationship between the three sides of a right triangle (a triangle with a 90^o angle).

Pythagorean theorem states that in a right triangle, the area of the square formed by the hypotenuse is equal to the sum of the areas of the squares formed by the other two sides.

The hypotenuse is the longest side in a right triangle, and it's always the side opposite to the 90^o angle.

Diagram showing the hypotenuse in a right triangle — The hypotenuse in a right triangle

Pythagorean Theorem Formula

a 2 + b 2 = c 2

Using some algebra we can derive the following equations to find the length of any side of a right triangle.

Hypotenuse c =
Side a =
Side b =

Pythagorean Theorem Proof

Algebraic Proof

Diagram showing four right triangles arranged in a way to form a large square and an inner tilted square — Pythagorean theorem algebraic proof

Looking at the above diagram, we see four copies of the same right triangle arranged in a way that forms a large square with a side length of (a + b), and a smaller inner square with sides c.

We also know that the sum of the areas of all the smaller pieces, that is the area of the four triangles (4 * (ab / 2)), and the area of the smaller inner square c². must be equal to the area of the large containing square (a + b)².

So:

(a + b) * (a + b) = c² + 2ab

By expanding the brackets on the left-hand side we get:

a² + 2ab + b² = c² + 2ab

Subtracting (2ab) from both sides gives us the pythagorean theorem:

a² + b² = c²

Visual proof

Rearrangement proof of the pythagorean theorem

Pythagorean Theorem Applications

There are many applications for the pythagorean theorem, and it's used everywhere in mathematics, physics, and engineering fields, some of the most common use cases for the pythagorean theorem are:

To find the length of any side of a right triangle when the other two sides are known.
To find the distance between two points.
To find the diagonal of a square or rectangle.

Pythagorean Theorem Example

Find the length of the hypotenuse of a right triangle when the other two sides have a length of a = 4cm, and b = 6cm.

First write the pythagorean equation:

c² = a² + b²

Now let's substitute the given values in place of the variables:

c² = 4² + 6²

Squaring the numbers gives us:

c² = 16 + 36 = 52

Now take the square root of both sides to get the length of the hypotenuse:

Is pythagorean theorem only for right triangles?: Yes, the pythagorean theorem is a special case of the law of cosines, and it only works for right triangles.

What is pythagorean theorem in math?: Mathematically it's $c 2 = a 2 + b 2$ , where c is the hypotenuse of a right triangle, a and b are the other two sides.

Who uses pythagorean theorem?: The pythagorean theorem is most frequently used by, mathematicians, physicists, engineers, construction related work, and many more.