The Pythagorean Theorem has been used in many branches of mathematics as well as various places outside of mathematics. This theorem provides a significant union between geometry and algebra as the foundation of the distance formula. In geometry the Pythagorean Theorem can help you find the length of a missing side and if slightly altered, the formula for finding the distance between two points is revealed.

Pythagoras' Theorem reads,

a^{2} + b^{2} = c^{2} .

The side lengths, a and b, are replaced with the distance between the x-coordinates of two point and the y-coordinates of those points. The hypotenuse, c, is equal to the distance between those points, represented by d.

(x_{1} - x_{2})^{2} + (y_{1} - y_{2})^{2} = d^{2}

To find d, the square root of both sides is taken. The result is the distance formula.

d = (x_{1} - x_{2})^{2} + (y_{1} - y_{2})^{2}.

Trigonometry, another branch of mathematics, relies heavily on the Pythagorean Theorem. The elements of trigonometry are based on sides of a right triangle, therefore connecting the geometric theorem to this offshoot of mathematics. In fact, the identities used in trigonometry are often referred to as the Pythagorean identities. The trigonometric identities spoken of are the following:

cos^{2} + sin^{2} = 1, 1 + tan^{2} = sec^{2}, and cot^{2} + 1 = csc^{2} (Agarwal).

The format these identities follow is equivalent to that of the Pythagorean Theorem, noting that 1^{2} = 1.

Science has adopted the Pythagorean Theorem in nearly every branch. Physics in particular has many applications of the theorem. If generalized to three dimensions, the Pythagorean Theorem can be applied to small intervals of distance to describe curved shapes, which is applying it to calculus. Following this idea, Einstein discovered a four dimensional form of the theorem which led to his exploration of his special theory of relativity. Later, he expanded it further for his study of the general theory of relativity (Agarwal). The equation derived for relativity also follows the pattern of the Pythagorean Theorem:

0 = E^{2} - (cp)^{2} - (mc^{2})^{2} (Overduin).

In an article on the Pythagorean Theorem, Agarwal states, "It is of vital importance in problems ranging from carpentry and navigation to astronomy." There are many uses for the famous theorem in the real world, the idea weaves its way into architecture, sports, music, art, and other practical situations of life. For example, if you need to use a ladder to change a lightbulb and there is a certain distance the ladder will stand away from the wall, you would need the Pythagorean Theorem to know how tall the ladder must be to reach the desired height. If you want a new TV that will fit in an entertainment center, you can maximize the screen size by measuring the space and calculating the hypotenuse using Pythagoras' Theorem to find the desired television size. Numerous examples and situations can be developed which show the application and significance of this theorem in everyday life.

There are also very specific uses for the Pythagorean Theorem in different career fields. The special case Pythagorean triple, 3-4-5, is commonly used in cabinetry and woodworking. They also use the theorem to calculate other lengths in projects and design. Similarly, architects rely on the Pythagorean Theorem to calculate heights of buildings and walls. Engineers and astronomers need the formula to solve for various distances in flight paths of spacecraft or satellites. There are many other professions where a knowledge and understanding of the Pythagorean Theorem is crucial to the success of the job.

Many sports use applications of the Pythagorean Theorem as well. Football has many instances where right triangles are involved, which relates the Pythagorean Theorem to the game. Receivers can calculate how far they need to run in a route to get to open space, or how far they ran to make a catch in previous plays. On defense, the players want to use the quickest route to the ball to make a tackle. This is measured as the hypotenuse of a right triangle as well. The following YouTube video from the National Science Foundation gives an explanation from the NFL on how the Pythagorean Theorem can calculate the desired "angle of pursuit" defenders should take to catch their opponent and make the tackle (The Pythagorean Theorem - Science of NFL Football, n.d.).