# Question #7a536

May 29, 2017

if the circumference is $9 \pi$, then $A \approx 3.53$ , but if the area of the whole circle is $9 \pi$, then $A = \frac{\pi}{2}$

#### Explanation:

So I'm asuming you have a circle with a circumference of $9 \pi$ in wich you want the area of a sector?

If the angle that forms the sector is given in radians, then we can use the formula: $A = \frac{1}{2} {r}^{2} \theta$
In this case, $\theta = \frac{1}{9} \pi$

We don't know the radius, therefore we must find it first.
The circumference of a circle is given by: $O = 2 r \pi$
Therefore:
$9 \pi = 2 r \pi \iff 9 = 2 r \iff r = 4.5$

Now we can calculate the area of the sector: $A = \frac{1}{2} \cdot {4.5}^{2} \cdot \frac{1}{9} \pi \approx 3.53$

... Or maybe, you meant the are of the whole circle is $9 \pi$ in many ways this would make more sense.

The solution is found in a similar way, only we will have to define two areas. The area for the whole circle and the area of the sector:
Aw is the area of the Whole circle.
${A}_{\text{w}} = {r}^{2} \pi = 9 \pi$

Isolate "r".

${r}^{2} \pi = 9 \pi \iff {r}^{2} = 9 \iff r = \sqrt{9} = 3$

Now calculate the area of the sector in the same way we did before:

${A}_{\text{s}} = \frac{1}{2} \cdot {3}^{2} \cdot \frac{1}{9} \pi = \frac{\pi}{2}$

This answer looks much better, which is why I think this is more correct.

Sorry for the inconvienience