What is the area of the shaded region in the given circle in terms of pi and in the simplest form?

Question

deercult · Answer

attachment

abdullahm · Answer

We have to find the area of the sector and subtract it by the area of the equilateral triangle.

That way we find the area of the unshaded region.

We then find the area of the circle and subtract the area of the unshaded region from it to get your final answer.

Ready to begin?

deercult · Answer

Ok!
Yep! I'll do my best to follow along!

abdullahm · Answer

To find the area of the sector:

$\Large\bf Area =\frac{n}{360}\pi r^2$

where n is the angle. In our case n is going to be 60. So plug it in and find the area of the sector. Area of a sector basically means the area of a slice of the circle.

deercult · Answer

$$\frac{ 60 }{ 360 }\pi6^{2}$$

Is this the right equation? If so, then you'd get,

$$0.2\pi36 $$

Which ends w/ 22.6
Is that right?

abdullahm · Answer

Well

$\Large\frac{1}{6}(6^2) \pi$

$\Large =6\pi\approx 18.85$

abdullahm · Answer

Now to find the area of an equilateral triangle, we can use this formula:

$\Large\bf Area = \frac{\sqrt{3}}{4}a^2$

where a is the side of the triangle

In our case a = 6

deercult · Answer

Ohh. Alright, one second!

deercult · Answer

Is it 15.6?

deercult · Answer

Or 15.59

abdullahm · Answer

Let's go with 15.59 since that's more accurate!

abdullahm · Answer

Now, what is

18.85 - 15.59 = ?

deercult · Answer

Ok!
I got 3.26!

abdullahm · Answer

Correct!

Now we need to find the area of the circle

$\Large\bf Area = \pi r^2$

where r is the radius, and r is 6 in our case.

What is the area?

deercult · Answer

$$36\pi$$

Or 113.04

abdullahm · Answer

and then subtract 3.26 from that!

abdullahm · Answer

113.04 - 3.26 = ?

deercult · Answer

109.78!

abdullahm · Answer

That's your final answer!

deercult · Answer

Ok! One thing, the answers are all in this format: http://prntscr.com/b0qn0o

deercult · Answer

Sorry D: I only juts noticed

abdullahm · Answer

Oh, they wanted an exact answer.

So let's just put all of our work together in the exact answers.

$\Large\bf 36\pi - (6\pi - 9\sqrt{3})$

That's basically what we did. And if we simplify that, we get:

$\Large\bf 36\pi - 6\pi + 9\sqrt{3}$

which is the same thing as

$\Large\bf 30\pi + 9\sqrt{3}$

Do you understand? $\ddot\smile$

deercult · Answer

Ah, yeah, I understand! Thanks a ton!

deercult · Answer

Thank you so much!!!

abdullahm · Answer

It was my pleasure!