Let me (hopefully) shed some light on what's going on here with the help of a picture:

[attachment=0:2ffyl8kh]480x320_Annular Sector.jpg[/attachment:2ffyl8kh]

As you can see we have your Annular Sector there, a thin slice of a "doughnut".

**Area:**
So your area would be the space inside the slice there, which daon helpfully provided:

[tex]A = {R^2 - r^2 \over 2}\theta[/tex]

Which - in English - is the difference of the larger circle's radius and the inner circle forming the "doughnut", then the [tex]\theta[/tex](pronounced "theta" and written using TeX, like this: '\theta' [minus the quotes]) there is the

fraction of the complete "doughnut" you're asking about (54 degrees in this case). But! Since we're talking about

degrees instead of

radians, we have to

convert between the two in order to get the right answer. Which is where all this talk about [tex]\theta={\pi\theta^{\circ}\over180}[/tex] comes from (Where [tex]\theta[/tex] is radians and [tex]\theta^{\circ}[/tex] is the number of degrees you have).

**Perimeter:**
So the perimeter (as per the figure) is:

[tex]P_{total} = a+b+P_1+P_2[/tex]

[tex]a[/tex] and [tex]b[/tex] are easy, they're just difference between [tex]R[/tex] and [tex]r[/tex]; And there are two of them so we see where the:

[tex]a+b = 2(R - r)[/tex]

Came from.

The curved portions [tex]P_1[/tex] and [tex]P_2[/tex] are a bit harder. But we can find them easily with the help of

arc length. Which is [tex]L = \theta r[/tex], where [tex]\theta[/tex] is the portion of the circle we want to measure and [tex]r[/tex] is the radius. Since we're talking about degrees instead of radians in this case, we use the conversion already mentioned:

[tex]\theta={\pi\theta^{\circ}\over180}[/tex]

So our formula ends up being:

[tex]L_\circ = {\pi\theta^{\circ} r \over 180}[/tex]

There are two lengths there with differing radii, so that's where Daon pulls the second part of the

equation from:

[tex]P_1+P_2 = {\pi\theta^{\circ} \over 180}{(R +r)}[/tex]

You need to add the straight edges (indicated by a & b - in your picture) for perimeter (daon's final answer includes that)
Hope that's clear.

--Pontifex

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