# Can anyone help solve this area between two circles problem, please?

There are 2 circles, circle A and circle D. They overlap at the centers. Segment AD connects the two centers and is 10 inches long. Segment AD is a side to square ABCD. What is the area of the region at the top of square ABCD, ending at the top of the circles?

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• There are 2 circles, circle A and circle D.

They overlap at the centers.

Segment AD connects the two centers and is 10 inches long.

Segment AD is a side to square ABCD.

What is the area of the region at the top of square ABCD,

ending at the top of the circles?

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• Let me see if I'm understanding your description correctly:

ABCD is a square whose sides have length 10. A circle centered at A contains points B and D. Another circle centered at D contains points A and C.

First, notice that square ABCD contains a quarter of each circle (or in correct terminology, a 90° sector). You know the radius of the circles, so the area of one sector is easy to calculate. The problem is that the sectors of the two circles overlap. We need to find the area of that overlap.

To start, calculate the area of the 90° sector. I'll refer to that area as X.

Now consider the point E where the two circle intersect inside ABCD.

AE = 10 and DE = 10, so AED is an equilateral triangle.

Calculate the area of triangle AED. I'll call that area Y.

Now consider the sector of circle A that follows the arc from E to D. This sector has the same 60° angle as the triangle, so it is 1/6 of the circle.

Calculate the area of this sector, which I'll call Z.

Inside the sector but outside the triangle is a segment of area Z-Y.

The overlap of the two circles is one triangle and two segments, or equivalently one 60° sector and one segment, or also equivalently two sectors minus one triangle.

Thus the area of the overlap is 2Z - Y.

Likewise, the part of the square taken up by the two quarter-circles is 2X - (2Z-Y).

And finally, the region at the top has area 100 - 2X + 2Z - Y.

• Thank you!

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• Anonymous
1 month ago

What does 'overlap at the centres' mean?

Are A and D, points or circles?  - you said they are both.

Needs a diagram

• I'm sorry, I tried uploading a photo as a PNG and then as a JPEG file, but it said "There was a problem. Try again later." for both. So I just tried to describe it as best as I could.

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