Question

Find the perimeter and area of this figure:
A square with a diagonal length of 18 meters.

I got the perimeter of 36squarerootof2, which the book said is right, but i'm not getting the area of it right. The book said the area is 162 meters squared, i keep getting 81

Answer 1

well, as you have probably figured that every side of your square is \(\normalsize\color{royalblue}{ 9\sqrt{2} }\), because the diagonal of a square in relation to the square's side, is exactly the same as the hypotenuse in a \(\normalsize\color{royalblue}{ 45-45-90{\tiny~}~{\rm degree~~triangle }}\) in relation to one of the legs of this triangle.

So, yes, the perimeter will hence be:
\(\normalsize\color{royalblue}{ P=4\times( 9\sqrt{2})=36\sqrt{2} }\)
\(\normalsize\color{royablue}{ (}\)Since, to find the perimeter of a square, knowing it's 1 side, you would just multiply the side times 4, because all sides of a square are same\(\normalsize\color{royalble}{ ) }\)

The area of a square is `side times side` (or ` (side) ^2 ` )
In your case it is going to be:
\(\normalsize\color{royalblue}{ A=(9~\sqrt{2}~)^2 }\)

To simplify this:
\(\normalsize\color{royalblue}{ A=(9~\sqrt{2}~)^2 }\)
\(\normalsize\color{royalblue}{ A=(9)^2\times~(\sqrt{2}~)^2 }\)
\(\normalsize\color{royalblue}{ A=~? }\) (you tell me)

Answer 2

and just in case, but I think you probably know, that:

For any number 'W' the following is true: \(\normalsize\color{blue}{ \left(\sqrt{{\color{white}{\large |}}{\rm W}} ~\right)^2={\rm W}}\)

Answer 3

good luck!
If you have any questions, please make sure to ask!