# circle

• Sep 18th 2008, 08:17 AM
fabxx
circle
In the figure below, QR is the arc of a circle with center P. If the length of arc QR is 6pi (i don't know how to do the pi sign, the 3.14 etc), what is the area of sector PQR?

If you can't see the attactment clearly, angle QPR is 30 degrees and the arc opposite of QPR is 6pi. I don't get why the correct answer is 108pi

Thanks a lot!!!!!!!!
• Sep 18th 2008, 08:21 AM
james_bond
Attaching the figure would be nice of you.
• Sep 18th 2008, 09:34 AM
masters
Quote:

Originally Posted by fabxx
In the figure below, QR is the arc of a circle with center P. If the length of arc QR is 6pi (i don't know how to do the pi sign, the 3.14 etc), what is the area of sector PQR?

If you can't see the attactment clearly, angle QPR is 30 degrees and the arc opposite of QPR is 6pi. I don't get why the correct answer is 108pi

Thanks a lot!!!!!!!!

Circumference:
$C=2 \pi r$

Area:
$A= \pi r^2$

The sector in question is $\frac{30}{360}=\frac{1}{12}$ of the circle. So the area of the sector will be $\frac{1}{12}$ of the area of the circle.

Problem is, we don't know the radius yet.

The arc length $6 \pi$ is also $\frac{1}{12}$ of the circumference, so the circumference must be $12 \times 6 \pi = 72 \pi$

Substituting into the circumference formula:

$2 \pi r = 72 \pi$
$r=36$

As stated before, the area of the sector is 1/12 the area of the whole circle, so:

Area of sector = $\frac{1}{12}\times \pi \times 36^2=\boxed{108 \pi}$