My Math Forum Circumference of smaller half circle removed inside of half circle?

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 September 10th, 2018, 09:56 AM #1 Member   Joined: Sep 2014 From: Sweden Posts: 94 Thanks: 0 Circumference of smaller half circle removed inside of half circle? So I was trying to find the circumference of a half circle inside of a half circle. By that I mean that I have an empty half circle inside of the larger half circle. So I will need to subtract the smaller half circle at the end. Circumference of half circle: $\displaystyle (\pi * d) / 2$ So assuming what I know is that $\displaystyle d1 = x + 2 + x = 2x + 2$ and that one part in the middle is missing so I only need to add $\displaystyle 2x$. So the equation for O1 (circumference 1) will be: $\displaystyle ((\pi * (2x + 2)) / 2) + 2x$ The diameter of the inner half circle is 2 so therefore: $\displaystyle O2 = ((\pi * 2) / 2) + 2$ $\displaystyle O2 = \pi + 2$ Then I need to subtract O1 with O2 to get the correct circumference. Let's call this O. $\displaystyle O = O1 - O2$ $\displaystyle O = \pi * (x + 1) + 2x - (\pi + 2)$ $\displaystyle O = \pi * (x + 1) + 2x - \pi - 2$ $\displaystyle O = \pi * x + \pi + 2x - \pi - 2$ $\displaystyle O = x(\pi + 2) - 2$ So this is what I get and I don't see where my mistake is. The book's answer is: $\displaystyle (\pi +2)x + 2 * \pi$ Last edited by skipjack; September 10th, 2018 at 03:06 PM.
 September 10th, 2018, 12:57 PM #2 Math Team   Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1038 What's that link got to do with the problem?
 September 10th, 2018, 03:30 PM #3 Global Moderator   Joined: Dec 2006 Posts: 20,942 Thanks: 2210 What does $x$ refer to? There's no reason to subtract $O2$ from $O1$. Just add $2x$ to the sum of the lengths of the curved lines: $\pi(x + 1) + \pi + 2x = (\pi + 2)x + 2\pi$.
 September 14th, 2018, 08:59 AM #4 Member   Joined: Sep 2014 From: Sweden Posts: 94 Thanks: 0 So $\displaystyle \pi$ is the length of the inner half circle's curved line? And then we add the bottom lines of the outer half circle $\displaystyle x + x = 2x$? So what we are doing is taking the curved lines of the outer and inner circle and adding them. Then we add the bottom lines to get the whole circumference. $\displaystyle x$ refers to the bottom lines at the bottom to the left and right. Forgot to specify that. Last edited by DecoratorFawn82; September 14th, 2018 at 09:03 AM.

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