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September 10th, 2018, 09:56 AM   #1
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Question 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.
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September 10th, 2018, 12:57 PM   #2
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What's that link got to do with the problem?
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September 10th, 2018, 03:30 PM   #3
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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: $\pi(x + 1) + \pi + 2x = (\pi + 2)x + 2\pi$.
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September 14th, 2018, 08:59 AM   #4
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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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