Area Common to 2 Circles: Radians Question

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In summary, the conversation is discussing how to find the area common to two circles with radii of 5cm and 12cm, which are partly overlapping with centers 13cm apart. The suggested approach involves drawing a triangle connecting a point of intersection and the two centers, and using the law of cosines to find the angle formed by the line connecting the centers. However, due to the side lengths of the triangle, it is easier to find the angle using sine instead.
  • #1
david18
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"two circles of radii 5cm and 12cm are drawn, partly overlapping. Their centres are 13cm apart. Find the area common to the 2 circles"

I'm not quite sure how to do this. I think I am meant to be using the area of sector as if I draw a line down the middle of the area formed I can use the 1/2r^2(x-sinx) but I don't know how I can find the angle of the triangle that is formed.

Any help?
 
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  • #2
Try drawing a triangle connecting a point of intersection of the two circles and the two centers of the two circles. Technically, you can then find the angle formed by the line connecting the two centers with one of the other sides of the triangle using the law of cosines, but because the side lengths of the resulting triangle are 5-12-13, it is much easier to find sin x.
 
  • #3
Partly overlapping is INTERSECTING.
 

1. What is the formula for finding the area common to 2 circles in radians?

The formula for finding the area common to 2 circles in radians is A = r²(θ - sinθ), where r is the radius of the circles and θ is the central angle in radians.

2. How do you convert degrees to radians?

To convert degrees to radians, you can use the formula π/180 * θ, where θ is the angle in degrees. For example, to convert 45 degrees to radians, you would use π/180 * 45 = π/4 radians.

3. Can you explain why radians are used to measure central angles?

Radians are used to measure central angles because they provide a more direct and consistent relationship between the angle and the arc length. This makes it easier to use in calculations involving circles and also eliminates the need for unit conversions.

4. How do you find the area common to 2 circles when the radii are different?

If the radii are different, you can still use the same formula A = r²(θ - sinθ), but you will need to find the central angle θ using trigonometric functions. Alternatively, you can split the area into smaller sections and add them together.

5. Can you find the area common to 2 circles if the circles overlap completely?

No, in this case, the area common to 2 circles would be the same as the area of one circle. This is because the entire area of one circle is shared by the other circle, so the overlapping area is equal to the total area of one circle.

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