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AOB is a quadrant sector of radius r cut from a circle of center O. A semicircle of center P is drawn outside of quadrant taking AB as diameter. What is the difference between area of whole geometry and that of quadrant?

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I want the area of ACBD (the shaded part). I proceeded as follows:

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What next??

Shubham Kumar Verma
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  • The entire circle with diameter $AB$ is outside the quadrant, unless you count the intersection with $O$ as "inside" the quadrant. And "whole geometry" is what? The entire circle plus semicircle? I would guess not, but we should not have to guess. – David K Sep 02 '19 at 19:22
  • I have updated the question. Please help. – Shubham Kumar Verma Sep 03 '19 at 07:03
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    The diagram makes everything much clearer. The work you have done so far also is a big improvement to the question. Next time you might use MathJax for the formulas (https://math.stackexchange.com/help/notation), but I have tried to answer this question based on the new information. – David K Sep 03 '19 at 10:10

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I agree with your calculations for the area of the quadrant sector and for the areas of the two parts in which the sector is cut by the chord $AB,$ namely, the triangle $\triangle AOB$ and the remaining part $ADBP,$ which is called a circular segment. I also agree about the radius of the semicircle.

The shaded region $ACBD$ is a type of figure called a lune.

From the radius of the semicircle you should be able to find its area, which is the combined areas of the circular segment $ADBP$ and the lune $ACBD.$ Then, knowing the area of the circular segment, you should be able to find the area of the lune easily.

This is actually a rather famous result in classical geometry.

David K
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