SOLUTION: a square and a semicurcular region have the same perimeter. If the length of the radius of the semicirular region is 8 what is the length of a side of a square?

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Question 223686: a square and a semicurcular region have the same perimeter. If the length of the radius of the semicirular region is 8 what is the length of a side of a square?
Found 2 solutions by tutor_paul, rapaljer:
Answer by tutor_paul(519) About Me  (Show Source):
You can put this solution on YOUR website!
Perimeter of a semicircle is given by:
p=%28pi%29r%2Bd
In this case, you are given r, so the perimeter of the semicircle (and the square) is:
p=8%28pi%29%2B16
The equation for the perimeter of a square (s=side of the square) is:
p=4s%29
Since the perimeter of the square is the same as the circle, you can substitute the expression found for the perimeter of the circle into the equation for the perimeter of the square and solve for s:
8%28pi%29%2B16=4s
highlight%28s=2%28pi%29%2B4%29
====================
Good Luck,
tutor_paul@yahoo.com

Answer by rapaljer(4671) About Me  (Show Source):
You can put this solution on YOUR website!
If the radius of the semicircle is 8, then the circumference of the entire circle is C=2%2Api%2Ar=2%2Api%2A8.

Since it's a semicircle, take half of this, and add in a diameter of 16, and you get the perimeter of the semicircle is pi%2A8%2B16

Let x = side of the square, which has a perimeter of 4x.

Since the perimeter of the square equals the perimeter of the semicircle,
4x=8%2Api%2B16

Divide both sides by 4:
x=2%2Api+%2B4

R^2

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida
Altamonte Springs Campus