Math: geometry question

[joonkimdds]

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There are two circles with shaded area, A and B.
Radius of B is twice of the radius of A.
angle of B is twice of the angle of A.
If the area of the shaded region of A is 3, what's the area of the shaded region of B?

answer is 24.

Could someone help me?

 

[pbure9]

There are two circles with shaded area, A and B.
Radius of B is twice of the radius of A.
angle of B is twice of the angle of A.
If the area of the shaded region of A is 3, what's the area of the shaded region of B?

answer is 24.

Could someone help me?

I got 6 the first time I looked at it too. LOL

I'll solve it out, and I think you'll see where the two of us made the mistake:

Area of first circle is r^2*pi
The shaded region however, should be seen as a fraction of 360. So just say theta/360. Theta will be "&" for this example 😉

So the area for A of the shaded region is r^2*pi*(&/360)

B, the area is (2r)^2*pi. Now the angle is 2&.
So the area of the shaded region will be (2r)^2*pi*(2&/360) or 4r^2*pi*(2&/360).

Now you know that A, r^2*pi*(&/360), equals 3

You can factor that out of our formula since 4 and 2 are both on the numerator. It looks as follows:
4r^2*pi*(2&/360) = 4*2*r^2*pi*(&/360) = 4*2*3 = 24
😀

 

[joonkimdds]

Aha~!

 

[Streetwolf]

A much easier way...

The angle is linear in nature so that just doubles the area.
The radius deals with area which is 2 dimensional. Thus by doubling something in 2 dimensions, you are increasing area x4. (Remember that the area of a circle uses r^2 and not just r.)

Thus increasing by x2 and x4 = increasing by x8.

3 x 8 = 24.

 

[pbure9]

A much easier way...

The angle is linear in nature so that just doubles the area.
The radius deals with area which is 2 dimensional. Thus by doubling something in 2 dimensions, you are increasing area x4. (Remember that the area of a circle uses r^2 and not just r.)

Thus increasing by x2 and x4 = increasing by x8.

3 x 8 = 24.

I would have never seen that, but that is why you are the math maser 😛
Good stuff man

 

[jay47]

A much easier way...

The angle is linear in nature so that just doubles the area.
The radius deals with area which is 2 dimensional. Thus by doubling something in 2 dimensions, you are increasing area x4. (Remember that the area of a circle uses r^2 and not just r.)

Thus increasing by x2 and x4 = increasing by x8.

3 x 8 = 24.

Exactly my first thought. Just did some math questions today with my Kaplan class that dealt with similar triangles and area. Same concept applies: doubling the length of one side, and keeping other sides similar, quadruples the area. Or 2^2.
The good stuff comes when you have three dimensions.....