Pythagorean Theorem

Applying and Visualizing the Pythagorean Theorem

The Pythagorean Theorem is a theory that states the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides. It is a fundamental principle in geometry that relates the sides of a right-angled triangle. The Pythagorean Theorem is used in real-life applications throughout different fields. The most obvious are architecture and construction to accurately plan and build structures, focusing on corners and support systems. Another field that utilizes the Pythagorean Theorem is in technology when creating applications and computer graphics of three-dimensional designs. More recently, in fields involving game development and medical imaging, the particularity of the Pythagorean Theorem is critical to calculate spatial relationships and dimensional distances.


The Pythagorean Theorem can be mathematically written as follows:

c = hypotenuse

a and b = the sides of the right triangle

Pythagorean Theorem


The equation can be used to find any side given two sides.

Example 1

right triangle with side lengths 5 and 18

             Plug inside values into the a and b sections                           52 + 182 = c2

             Simplify a and b                                                                           25 + 324 = c2

             Add a and b                                                                                   349 = c2

                Take the square root of both sides                                           √349 = c

             Take the square root of 349                                                       18.68 = c  


The value of the hypotenuse is 18.68. The Pythagorean Theorem can be used no matter what two values were given, including the hypotenuse.

Example 2

right triangle with side length 4 and hypotenuse length 6


            Plug in the values in the Pythagorean Theorem                42 + b2 = 62

            Simplify a and c                                                                         16 + b2 = 36

            Subtract 16 on both sides to isolate b                                   b2 = 20

            Take the square root of both sides                                        √b2 = Ö20

            Simplify                                                                                       b = 4.47


The Pythagorean Theorem can be required to solve word problems. Read the following word problem and solving using the Pythagorean Theorem.

Example 3

A flagpole in the school courtyard is 20 feet high and casts a 7-foot shadow. What is the distance between the top of the flagpole to the top of its shadow cast.

right triangle with side lengths 7ft and 20ft


Drawing an image will help map out what the word problem is asking for. You need to find the distance from the top of the flagpole to the end of its shadow cast. The distance will act as the hypotenuse. Begin to plug in your values:


            Set up the equation                                                     72 + 202 = c2

            Simplify exponents                                                     49 + 400 = c2

            Add                                                                                 449 = c2

            Square root both sides to isolate c                           √449 = c

            Simplify                                                                          21.19 feet


In conclusion, the Pythagorean Theorem has geometric concept that can be applied to real-life problems. Knowing this fundamental equation for right triangles, students, mathematicians, engineers, and others can calculate the side lengths of triangles.

Author: Maerie Morales