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Need Help For Area of Circle Segment

Attachment 24065

Problem:

Two concentric circles are shown in the figure above. The chord AB of the larger circle

is tangent to the smaller circle. The area of the larger circle is twice the area of the smaller circle. What is the area of the segment of the large circle (shown in dashed lines) that is cut off by the line segment AB? (Gee, I wish the image could be larger)

Inasmuch as the radii are not given directly nor the central angle, it seems that the radius must be expressed as a function of the areas of the circle. Since the larger circles area is twice the smaller circle’s, I expressed the smaller circle’s radius as r = sq.root of A÷ pi and the larger circle’s radius as r = sq. root 2A÷ pi. The height of the triangle portion of the sector would be the radius of the smaller circle.

I would like to use the formula for the area of the segment directly, i.e. Area = ½ r^{2 }(pi ÷ 180 * Ø – sin Ø), (for theta in degrees), but am at a loss as to how to do this without a very messy equation and not having the central angle. To solve for the sector area and substract the triangle’s area to obtain the segment area also eludes me. It seems all the triangles sides would be written in terms of something else, i.e. the larger circle’s radius.. Any help will be appreciated in how to proceed with this. Obviously, I am missing something here.

Re: Need Help For Area of Circle Segment

given ...

$\displaystyle \pi R^2 = 2\pi r^2$

solving for $\displaystyle R$ ...

$\displaystyle R = \sqrt{2} \cdot r$

this relationship between $\displaystyle R$ and $\displaystyle r$ says that the central angle is $\displaystyle 90^\circ$

segment area ...

$\displaystyle A = \frac{\pi R^2}{4} - \frac{R^2}{2} = \frac{R^2(\pi - 2)}{4}$

Re: Need Help For Area of Circle Segment

Skeeter, thanks so much for your help. I do have a question for my clarification.

In your given equality: pi* R^2 = 2* pi * r^2. First, is the large R the radius for the smaller circle? If so, it appears that the two areas are being equated. I’m confused about this inasmuch as one is twice the other. Can you elaborate?

Also, in the calculation for the segment area, the first portion of this formula is the area for a quarter circle sector, but I am confused about the term: minus R^2 ÷ 2. Can you elaborate on this term, please? Thanks again

Re: Need Help For Area of Circle Segment

R is the radius of the larger circle ... r is the small circle radius

you were given the fact that the large circle area is twice the small circle area

$\displaystyle \color{red}{\pi R^2} = \color{blue}{2 \pi r^2}$