Question

what is the ratio of the perimeter of a semicircular region to the perimeter of a circular region with the same radius?

Answer 1

I want to say that it is 1:2 (or 1/2).

Answer 2

the answer is 2+pi to 2 pi
i just dont know how to get to it

Answer 3

The only reason it's not 1:2 is because the semicircle also has a diameter included. See attachment.

Call the radius of each "r."
First, the semicircle:
The length of the diameter is 2r. You need to add this to half the circumference. The circumference of a circle is 2(pi)r, so half the circumference is pi(r). This means that the perimeter of the semicircle is
\[2r+\pi r\]Factor out a pi to get\[C_1=r(2+\pi)\]

Next, the circle:
The circumference is all we need here.
\[2\pi r\]Or alternatively,
\[r(2\pi)\]

Divide C1 by C2 to get your ratio.
\[r(2+\pi)/[r(2\pi)]\]Simplify to get your final ratio of
\[(2+\pi)/(2\pi)\]

Answer 4

Thanks, jabberwock; I didn't think to take the semicircle's diameter into consideration.

Answer 5

I thought that, too at first :(. Then she wrote the answer and I had to back-solve it.

Answer 6

Good work!