If you're studying geometry, you've definitely come across the famous equation: a² + b² = c².
This equation is better known as the Pythagorean Theorem. It is the fundamental rule that governs right-angled triangles. If you know the length of any two sides of a right triangle, this formula allows you to find the exact length of the missing third side.
Whether you're doing math homework, building a staircase, or calculating the diagonal size of a TV screen, this formula is incredibly useful.
What Does a² + b² = c² Mean?
To understand the formula, you first need to look at a right triangle (a triangle where one angle is exactly 90 degrees).
In this triangle, there are three sides:
- Side 'a' and Side 'b': These are the two shorter sides that form the 90-degree right angle. They are usually called the "legs" of the triangle.
- Side 'c': This is the longest side of the triangle, and it is always located directly across from the 90-degree angle. This side is called the hypotenuse.
The theorem states that if you square the length of leg a, and square the length of leg b, and add those two numbers together, the result will perfectly equal the square of the hypotenuse c.
The Classic Example: The 3-4-5 Triangle
Let’s look at the most famous example of this formula in action.
Imagine a triangle where the two legs are 3 inches and 4 inches long.
- a = 3
- b = 4
If we plug those into our equation:
- a² = 3 × 3 = 9
- b² = 4 × 4 = 16
- a² + b² = 9 + 16 = 25
Since a² + b² = c², we know that c² = 25.
To find c (the actual length of the hypotenuse), we just take the square root of 25.
- c = 5
This is known as a "Pythagorean triple" because all three sides are perfect whole numbers!
How to Find a Missing Leg (Finding a or b)
What if you already know the longest side (the hypotenuse c) and one of the legs (b), but you need to find the missing leg (a)?
You simply rearrange the formula!
Instead of: a² + b² = c²
You write: a² = c² - b²
Example:
Let's say your hypotenuse (c) is 13, and one leg (b) is 12.
- c² = 13 × 13 = 169
- b² = 12 × 12 = 144
- a² = 169 - 144 = 25
- Take the square root of 25 to find
a. - a = 5
The Easiest Way to Solve It
Doing square roots in your head isn't always easy, especially when the numbers aren't perfect whole numbers (for example, if your sides are 7.5 and 10.2).
If you don't want to do the math by hand, you can use our free a squared plus b squared equals c squared calculator.
How to use the calculator:
- Go to the Pythagorean Theorem Calculator.
- Select what you are trying to find (the Hypotenuse
c, or a missing legaorb). - Type in the two numbers you already know.
- The calculator instantly gives you the length of the missing side, complete with the step-by-step math shown right on your screen.
Understanding a² + b² = c² is one of the most practical math skills you can learn, and it forms the foundation for almost all advanced geometry and trigonometry!