5 Real-Life Applications of the Pythagorean Theorem

March 4, 2021

The Pythagorean theorem is an important theorem used in geometry that shows the relationship between the lengths of the sides of a right-angled triangle. It is named after the famous Greek mathematician Pythagoras and is also known as Pythagoras theorem. This theorem holds great significance in geometry and forms the base for solving various mathematical problems. It finds many real-life applications in the field of construction, navigation, surveying, architecture, etc.

What is Pythagoras theorem?

Pythagoras theorem states that the sum of the square of the two sides of a right-angle triangle is equal to the square of the third side called the hypotenuse. The equation formed as per the Pythagoras theorem is a^2 + b^2 = c^2, where a, b and c are the sides of a right triangle. The ability to find the length of a side when the other two sides’ length is given makes the Pythagorean Theorem a beneficial construction and navigation technique. Here are the five real-life applications of the Pythagorean theorem:


Pythagorean Theorem is applicable to calculate the length of the diagonal connecting any two straight lines. This aspect of the Pythagorean theorem is pretty useful in designing and construction. For instance, while building a sloped roof, if you know the height and the length of the roof to cover, you can use the Pythagorean Theorem to find the diagonal length or to calculate the roof’s slope. You can utilize this data to precisely cut beams to support the roof structure or to calculate the roof’s total area.

Painting Walls

Painters also make use of the Pythagoras theorem to paint buildings. They need to evaluate the ladder height and distance from the wall to complete the work without an accident. In this case, the ladder is the hypotenuse in terms of the theorem.


The Pythagorean theorem is useful in two-dimensional navigation, mostly used by ships to find the shortest routes. Sailors navigate using this theorem by making a horizontal and a vertical line from the current location to form a right triangle in order to find the shortest distance to the destination. The distances in each direction will be the two sides of the triangle, and the shortest line connecting them will be diagonal. The same law applies to other forms of navigation on land or the sky. For instance, a flight can use its altitude above sea level and its distance from the destination airport to find the exact geolocation to begin a safe descent to that airport.

Building Square & Angles

Right angles and square shapes are quite common in building designs and construction work. Engineers use the basic property of the Pythagoras theorem, which states that if the sides satisfy the theorem condition, the triangle will always form a right angle. While laying out the building foundation or constructing a right-angled corner between two walls, engineers set out a triangle from three strings that correspond with sides lengths satisfying the theorem.


The Pythagorean Theorem is used to calculate many aspects of the terrain that otherwise are hard to assess, such as the steepness of slopes of mountains. A surveyor uses a telescope and a measuring stick at a fixed distance away, so when the telescope’s line of sight and the measuring stick create a right angle, a triangle is formed. Since the surveyor has the information about the two sides of the triangle that are the height of the measuring stick and the horizontal distance of the stick from the telescope, he calculates the steepness of the hill by using the theorem to find the length of the slope that covers that distance.


Pythagorean theorem is foundational in various branches of mathematics, physics, geology, architecture, and more. There are numerous interactive activities for kids to reinforce the Pythagoras theorem’s understanding, like maths games, puzzles, and worksheets. Cuemath offers excellent learning resources for kids to gain a clear understanding of various concepts and their applications in our everyday life. Cuemath live online classes are designed to promote the real-life application of maths for kids to learn interestingly.

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