# Thread: geometry/algebra. Areas with only perimeter given

1. ## geometry/algebra. Areas with only perimeter given

I am having trouble figuring out where to begin.
The problem:

A figure has the shape of a square with semicircles on each side. The figure has a perimeter of 60ft. Find the area of the figure.

I am thinking that the arc of eac semicircle which would be that perimeter of each is 15 ft. But that is as far as I can get. I am frustrated. haha

Bill

2. Hello,

Originally Posted by ttlpkg32
I am having trouble figuring out where to begin.
The problem:

A figure has the shape of a square with semicircles on each side. The figure has a perimeter of 60ft. Find the area of the figure.

I am thinking that the arc of eac semicircle which would be that perimeter of each is 15 ft. But that is as far as I can get. I am frustrated. haha

Bill
For someone who doesn't know how to post, I find it wonderful that your title is so clear !

Ok, let's go for it.

Let x be the side of the square (see picture below).
Let's calculate the perimeter in respect with x.

We know that each semicircle is 15cm.

But ! You know that they are semicircles of a circle which diameter is x.
As the perimeter of a semicircle is half the perimeter of the circle :

$\displaystyle 15=\frac 12(2 \pi \frac x2)$

$\displaystyle 15=\frac{\pi x}{2}$

Hence $\displaystyle \boxed{x=\frac{30}{\pi}}$

~~~~~~~~~~~~

Now, the area !

The area is the sum of the areas of :
- the square
- the 4 semicircles.

You should know the area of the square

So let's see what the area of 1 semicircle is :

The area of a semicircle is half the area of the circle.
Hence $\displaystyle A_s=\frac 12 (\pi r^2)$, where r is actually $\displaystyle \frac x2$

---> $\displaystyle \boxed{A_s=\frac 12 (\pi \left(\frac x2 \right)^2)=\frac{\pi x^2}{8}}$

So what is the area of the whole thing ?