The figure below shows the top view of a circular room with a circular stage. The diameter of the stage is 24 feet. The shaded portion represents the seating for the audience around the stage.
What is the area of the seating portion?

Question

The figure below shows the top view of a circular room with a circular stage. The diameter of the stage is 24 feet. The shaded portion represents the seating for the audience around the stage. 
What is the area of the seating portion?

anonymous · Answer

attachment

jim_thompson5910 · Answer

Any ideas to get started?

anonymous · Answer

find the area of the cirlce in general?

jim_thompson5910 · Answer

That's a good first step

jim_thompson5910 · Answer

So what is the area of the larger circle?

anonymous · Answer

11304

jim_thompson5910 · Answer

how did you get that? I'm assuming you did pi*r^2 right?

anonymous · Answer

yeah thats what i did!

jim_thompson5910 · Answer

ok, It's probably better to leave it in terms of pi

jim_thompson5910 · Answer

So 

A = pi*r^2

A = pi*60^2

A = 3600pi

anonymous · Answer

alright...now what do i do?

jim_thompson5910 · Answer

now what is the area of the smaller circle?

anonymous · Answer

452.16 or  it be 144pi

jim_thompson5910 · Answer

144pi is what I'd go with

jim_thompson5910 · Answer

So the area of the outer ring is 

(Area of larger circle) - (Area of smaller circle) = 3600pi - 144pi = 3456pi

jim_thompson5910 · Answer

This is NOT the final answer as this includes that non-shaded bit in the outer ring.

anonymous · Answer

right. how do i find out how much of the circle is not shaded?

jim_thompson5910 · Answer

Now we turn to the formula for the area of a sector

Area = (angle/360)*pi*r^2

Does this formula look familiar?

anonymous · Answer

a little bit. ive seen way to many formulas haha

jim_thompson5910 · Answer

Basically, if you have a pie shaded sector (or pizza slice shaped sector) and you know the angle from the center that it makes like in the drawing below, then you can find the area of this slice

|dw:1341442006502:dw|

jim_thompson5910 · Answer

The angle of that pizza slice is x, so the area of that slice is then

A = (x/360)*pi*r^2

anonymous · Answer

which in this case would be (70/360)*pi*r^2...right?

jim_thompson5910 · Answer

You got it

jim_thompson5910 · Answer

You would just use different values of r

jim_thompson5910 · Answer

r = 60 for the larger slice and r = 12 for the smaller slice

jim_thompson5910 · Answer

the piece that is missing in the outer ring would then be equal to the difference in those areas

anonymous · Answer

alright!! i get it now!

jim_thompson5910 · Answer

alright great

jim_thompson5910 · Answer

tell me what you get for the final answer

anonymous · Answer

alright...i hope this is right. i got 1348

jim_thompson5910 · Answer

No it should be larger

jim_thompson5910 · Answer

What did you get for the areas of each pizza slice?

anonymous · Answer

i got 2196 and 87.92

jim_thompson5910 · Answer

Again I'd leave everything in terms of pi

anonymous · Answer

so 700pi and 28pi?

jim_thompson5910 · Answer

good

jim_thompson5910 · Answer

Their difference is then 700pi - 28pi = 672pi

jim_thompson5910 · Answer

So the missing piece in the outer ring (the unshaded bit in the outer ring) is 672pi

jim_thompson5910 · Answer

This means that the shaded area is then...

(Area of entire outer ring) - (Area of unshaded bit in outer ring)

(3456pi) - (672pi)

2784pi

anonymous · Answer

and thats with that piece missing?

jim_thompson5910 · Answer

yes that is the area of the shaded region as shown in the pic above

jim_thompson5910 · Answer

aka the original pic given to you

anonymous · Answer

right! i forgot to multiply it by pi...oops haha

jim_thompson5910 · Answer

If you want the approximate area, use pi = 3.14 to get 


2784pi = 2784*3.14 = 8741.76

jim_thompson5910 · Answer

lol that explains a lot

anonymous · Answer

no kidding. well AWESOME! thanks again so much! this really helped!

jim_thompson5910 · Answer

I'm glad that's all clicking for you