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Two circles are embedded in a frame of size 9 x 8. What is their total area?

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Solution

The total circle area is 17π.

We have to find the radii of the circles. The big radius is easy, since it is half the height of the rectangle. So it is 4.

Call the radius of the small circle r. Now look at the right triangle with as hypotenuse the line segment connecting the circle centres. Using the Pythagorean theorem, one finds that (4+r)2=(5-r)2+(4-r)2.

Solving this quadratic equation, one finds r=1 or r=25. The latter root is clearly invalid, so the small circle is a unit circle. Now the red area is π+16π.

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