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Puzzle #19

A white circle passes through the center/centre and the rightmost point of a yellow semicircle. A vertical arrow from the topmost point the semicircle to the bottommost point of the circle is labeled/labelled six.

The white circle passes through the centre of the yellow semicircle.

What is the area of the shaded shape?

Solution

Let rr and RR be the radii of the white circle and the yellow semicircle, respectively.

A white circle passes through the center/centre and the rightmost point of a yellow semicircle. A vertical arrow from the topmost point the semicircle to the bottommost point of the circle is labeled/labelled six. The radius of the semicircle, positioned vertically, is labeled/labelled R and the radius of the circle, positioned horizontally, is labeled/labelled r.

We can see in the diagram that R=2rR=2r and r+R=6r+R=6.

Then, r=2r=2 and R=4R=4.

So, the area of the shaded shape is

π×(4)22−π×(2)22\frac{\pi \times \left(4\right)^2}{2}-\frac{\pi \times \left(2\right)^2}{2}
8π−2π8\pi-2\pi
6π6\pi

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