Question

how do i find the area of the shaded area of the circle?

Answer 1

lol just wait till you get to the next one, thats the question I'm stuck on...Liberty U Online?

Answer 2

no, ORU eAcademy. I have half of the equation I think, but I need to figure out the rest.

Answer 3

It might help to divide the circle like so (see attached)

Answer 4

I know the area of the semi circle on the bottom is 72pi.

Answer 5

Jim is correct, it's the area of the whole circle MINUS the sum of two sectors (identical sectors, you can find one and double it thanks to symmetry) and a triangle.

\[Area_{total} = Area_{cicle}-Area_{section} = \Pi r^2 - (2*Area_{sector}+Area{\triangle})\]

Answer 6

ok, so the area of the circle is 144pi, but how do I find the area of the sectors? Primarily the triangle that Jim had made.

Answer 7

Do you have measurement values or is this just abstract variables?

Answer 8

Oh wait it's just got tinie-tiny #'s I see them now

Answer 9

That 12 is the radius I think, width of the section is 3, ok... one moment

Answer 10

well, there is the 3 unit height of the unshaded part, and the . with 12 above it, which I assume is the half way point of the circle meaning it has a 12 unit radius.

Answer 11

For a sector
\[Area_{sector}=\frac{\theta}{360^o}*\Pi r^2\]
as you can see we'll need the angle, which we can get by first finding out more information about Jim's triangle (for a name).

\[Area_{\triangle}=\frac{b*h}{2}\]
For me, and there's more than one way of doing this, I would cut that triangle in half down the middle and make two smaller right-angle triangles, each with a height of 3 units. I'll double those to find the area of the full triangle

Answer 12

FYI: that formula for a sector is in degrees, not radians

Answer 13

ok, i will try, but for that triangle, since its base is the smaller chord, how do I find the length of the chord?

Answer 14

The base is something formally called a "chord"

http://en.wikipedia.org/wiki/Chord_%28geometry%29

Check this link out :-)

Answer 15

In short, it's your Pythagorean theorem at work, along with your memory of SOH-CAH-TOA

Answer 16

ok.

Answer 17

Here, I'll give you the formula:
\[chord_{length}=2\sqrt{r^2+d^2}\]

In this case I believe r=6 and d=3

Answer 18

yeah. I think I got it. Thanks for the help.

Answer 19

\[ b=\frac{chord_{length}}{2}=\frac{(6\sqrt{3})}{2} \]

Are you able to find your final area based on the information now?

Answer 20

Yeah.

Answer 21

I would recommend leaving square roots and Pi until the end of your calculations, that way you don't have horrendous decimals

Answer 22

Yeah. The lesson wants me to leave pi as pi because they say its the most accurate.