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Area of a square from the diagonal

The properties of a square allows you to use the diagonal to find the area. 

A square has four equal sides and each corner forms a 90-degree angle.
The diagonal cuts each corner into two 45 degree angles.

As a result, you have a 45 45 90 triangle with the diagonal as the hypotenuse.

In order to find the area we need the length of at least one side.

The hypotenuse of a 45 45 90 triangle equals x√2 and the other two sides equal x 

We can use this information to find the area of a square.

For example, if the diagonal of a square equals 6 let's find the area. 

Set the diagonal equal to x√2 
6 =  √2
Divide both sides by √2 = 3√2
This equals x or the length of one side

Next, you can plug this into the area formula of a square
3√2 x 3√2 or Side x Side in order to find the area of the square.

3√2 x 3√2 = 18 
Area of a square from a diagonal of 6 = 18 square units