Pythagorean Theorem: what it is, examples and applications - Meanings

Pythagorean Theorem: what it is, examples and applications - Meanings
Posted on 09-02-2022

Pythagoras theorem

What is the Pythagorean theorem?

The Pythagorean theorem is a mathematical premise that allows us to calculate the length of the sides of a right triangle.

The statement of the Pythagorean theorem says the following:

In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs.

The formula to calculate the Pythagorean theorem is h² = a² + b².


The letters "a" and "b" represent the legs of the triangle, and the letter "h" the hypotenuse.

A right triangle is one that has an angle of 90º. The two sides that form the right angle are called legs, the opposite and longer side is called the hypotenuse.


The Pythagorean theorem is only applicable to this type of triangle and to perform the calculation we need to know the value of two of the sides of the triangle.

Examples of the Pythagorean theorem

Here is an example showing a right triangle for which we know the length of the legs, but we need to find the length of the hypotenuse. We are going to explain the process to find the value of the hypotenuse.


  1. We know that the legs of the triangle measure 6 and 8 respectively and we need to know the value of the hypotenuse (h).
  2. To do this, we square the value of the legs and add them to find out how much the square of the hypotenuse (h 2) is worth:
    2 = 6 2 + 8 2
    2 = (6x6) + (8x8)
    2 = 36 + 64 =
    100h 2 = 100
  3. We already know that h 2= 100, now we need to calculate the square root to know the value of h:

In the following example, we have a right triangle of which we know the measure of one of the legs and the value of the hypotenuse. Next, we explain the process to calculate the value of the unknown leg.


  1. We know that one of the legs measures 12 and the hypotenuse 13.
  2. To know the measure of side "b" we square the values:
    13 2= 12 2 + b 2
    (13x13) = (12x12) + b 2
    169 = 144 + b 2
  3. To find the value of b 2we have to subtract the square of the leg from the square of the hypotenuse:
    2 = 169 - 144 = 25
    2 = 25
  4. The result is b 2= 25, so we have to take the square root to find the value of b:
    b = √25
    b = 5.

Applications of the Pythagorean Theorem

A Pythagorean theorem is a basic tool in the study of trigonometry. It allows knowing the area of ​​any type of triangle through its division into right triangles.

Thanks to the Pythagorean theorem, the study of geometry was deepened, which represented an enormous advance for geographical knowledge and the elaboration of maps.

This premise is fundamental in architecture and engineering. It allows defining and calculating distances and measurements, necessary in construction projects.

The Pythagorean theorem is also used in forensic medicine to determine trajectories, in navigation and location systems to calculate location points, and in disciplines related to design, such as textile manufacturing.

Who was Pythagoras?


Pythagoras was an ancient Greek philosopher and mathematician, born on the island of Samos around 569 BC.

He completed his training in the main centers of knowledge of the ancient Mediterranean, Asia Minor, and Egypt. He was trained as a geometer priest, a doctrine that interprets the universe through numbers, and there he acquired his mathematical knowledge.

He founded the Pythagorean brotherhood in the city of Crotona. A scientific, philosophical, and spiritual school dedicated mainly to the study of numbers and geometry, although they also dealt with disciplines such as music and astronomy.

The brotherhood suffered an attack, as a consequence of its expansion and politicization, which forced Pythagoras to flee to Metaponto, where he died in 475 BC.


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