Semicircles/Square but only perimeter given

**ttlpkg32** · April 29th 2008, 10:01 AM

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

**Moo** · April 29th 2008, 10:10 AM

You didn't have to post again for correcting a mistake, you could use this :

**earboth** · April 29th 2008, 11:26 AM

Originally Posted by ttlpkg32

...

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.

...

1. Draw a sketch

2. The perimeter of the figure is:

$p = 2x+2 \pi \cdot \frac x2 = x(2+\pi)$

3. The perimeter is 60' :
$60 = x(2+\pi) ~\implies~ x=\frac{60}{2+\pi}$

4. The area of the figure is a square and a circle:

$A = x^2 + \pi \cdot \left(\frac x2\right)^2 = x^2\left(1+\frac{\pi}4\right)$

5. You should get: $A \approx 243.132\ squft$

**Mathnasium** · April 30th 2008, 10:29 AM

From the original post, sounds like it's a square with 4 semicircles attached - one on each side - not just two. Since the perimeter is 60, the OP's assumption that each semicircle has a perimeter of 15.

Now, take two of those semicircles, put them together, and you have a full circle with a perimeter of 30. Remember that $C = \pi d$ , so the diameter of each circle is $\frac{30}{\pi}$ - thus, the radius is $\frac{15}{\pi}$ - we'll need both the diameter and the radius for this.

Now, we have two full circles in our shape (4 semicircles), so they have a TOTAL area of $2\pi r^2$ . Putting in our radius gives:

$2 \pi (\frac{15}{\pi})^2 = \frac{450}{\pi}$ .

So that's the area of our four semicircles. Now, note that the square has a side length that's equal to the diameter of the circle, which we said was $\frac{30}{/pi}$ . To find the area of a square, just square the side length. So, the area of our square is:

$A = \frac{900}{\pi^2}$ .

To find the total area of both shapes, add them together:

$TA = \frac{450}{\pi} + \frac{900}{\pi^2} = \frac{450\pi + 900}{\pi^2} \approx 234.43$

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