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Using the following image for a visual: enter image description here

Is there a formula or equation I can use to find the area of the circle NOT overlapped with the rectangle (i.e. the filled in orange part)? I know all of the coordinates & sizes of both the rectangle and the circle, I'm just not sure how to apply them.

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What do u mean by having the coordinates? do u know the coordinates of the vertexes of the rectangle? do u know the coordinate of centre of the circle? – Babak S. Jun 6 '13 at 13:14
Sorry - wasn't specific enough. Yes, I know the x,y position of the rectangle, and the height/width. I also know the diameter, radius, and center of the circle. I can post them if you'd like, but I didn't want to seem like I'm trying to have someone solve it for me :) – lhan Jun 6 '13 at 13:15
Is it enough to do it when one side of the rectangle lies along the diameter of the circle, or could the rectangle and circle be placed in any way? – in_wolframAlpha_we_trust Jun 6 '13 at 13:15
they could be placed in any way – lhan Jun 6 '13 at 13:17
up vote 3 down vote accepted

You can break up the overlapping area into a segment and a triangle by drawing a diagonal line between the two intersection points and then the formulae for the areas of each of these shapes are very well known. Subtracting these areas from the area of the whole circle will then give you your answer.

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Thank you. I'll give this a try. – lhan Jun 6 '13 at 13:17
No problem at all. – john Jun 6 '13 at 13:18
Yep that worked out. Thanks again! – lhan Jun 6 '13 at 13:22
Will your method work if one of the sides of the rectangle does not pass through the center of the circle (as is presumably intended to be the case from the illustration included)? – Dilip Sarwate Jun 6 '13 at 14:56
Yes. It's all just a matter of geometry. Note that the method doesn't use the sides of the rectangle (except in the triangle area which is pretty basic), only the intersection points as these define the sector. – john Jun 6 '13 at 15:01

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