Pythagorean Theorem

Table of Contents

Introduction

The Equation

Applications

Pythagorean Triples

Conclusion

Introduction

The pythagorean theorem is probably the most known and most widely used equation in all of geometry. It gives a simple way to find the length of any side of a right triangle (provided you know the other two) using arithmetic.

The Equation

The Pythagorean theorem is one of the most commonly used formulas when dealing with triangles.

For a given right angled (90°) triangle, the Pythagorean theorem states that:

\(a^2+b^2=c^2\), where a and b are legs of the triangle and c is the hypotenuse (the longest side / the side directly opposite to the right angle)

Using this, the formula can also be manipulated so you can find the value for one side using:

\(c=\sqrt{a^2+b^2}\) → where you are trying to find the hypotenuse

\(a=\sqrt{c^2-b^2}\) → where you are trying to find the triangle’s leg

Applications

The Pythagorean theorem helps in 2 main ways:

It allows you to identify the missing side of the triangle if you know the remaining two sides as well as the fact that the triangle is right-angled

It helps you determine whether a triangle is right-angled if you know all 3 sides

Pythagorean Triples

Using the Pythagorean theorem, we can generate something called Pythagorean triples. Essentially, these are sets of 3 positive integers that fit the Pythagorean formula.>

Each set is in the form (a, b, c), where a2+b2=c2. A few common examples are:

(3, 4, 5)

(6, 8, 10)

(5, 12, 13)

(8, 15, 17)

Knowing a few of these Pythagorean triples are helpful as they will help you instantly recognize if a triangle is right angled, or allows you to easily find the remaining side of a right angled triangle with integer valued sides.