Find the area of a square with a diagonal of three centimeters.
Here, we have a square with a diagonal of three centimeters. And we want to find the area of a square. And the area of a square is equal to the length times the width. However, we don’t know the length or the width. But what can we say about the length and the width of a square?
Well, in a square, all sides are equal. So the length and the width would actually be the exact same measurement. So we could just call all of these sides 𝑥 because it’s the same value that’s missing for all four sides. Something else that we know about a square is that a square is made up of four 90-degree angles, which we’ve labeled.
Notice, however, this is a right triangle. And we can use that to find the length and the width. That way we can find the area. And we can use something called the Pythagorean theorem. And the Pythagorean theorem states that the square of the longest side is equal to the sum of the squares of the shorter sides. The longest side is always across from the 90-degree angle. So we can replace the longest side with three. And then the shorter sides are the other two sides. So we can replace those with 𝑥.
Now we simplify. Three squared is equal to nine. And 𝑥 squared plus 𝑥 squared is equal to two 𝑥 squared. So to solve for 𝑥 squared, we need to divide both sides of the equation by two. And we find that 𝑥 squared is equal to 4.5.
Now, before we square root this and solve for 𝑥, let’s think about what we said about the area. It’s the length times the width. But as we stated before, the length and the width are equal. They are both 𝑥. So the area is actually equal to 𝑥 times 𝑥 which is equal to 𝑥 squared. And we know what 𝑥 squared is equal to. 𝑥 squared is equal to 4.5.
Therefore, the area of the square is 4.5 centimeters squared.