Three stacked circles with different radii are inside a big circle, all tangent with each other. In the top it is the small circle, labeled/labelled pi. The medium circle is between the small circle and the bigger circle, it passes through the center/centre of the big circle and it is labeled/labelled four pi. The bigger circle is below the medium circle and it passes also through the center/centre of the big circle.

Three circles are inside of a big circle. The yellow circle and the red circle meet at the center/centre of the big circle.

What fraction of the big circle is painted?

Solution

As the red circle passes through the center/centre of the big circle, and the area of the others circles are π\pi and 4π4\pi, we get that its radius is 3.

Then, the radius of the big circle is the double, i.e., 6. So, the area of the big circle is 36π36\pi.

So, the fraction of the big circle that is colored is

π+4π+9π36π\frac{\pi+4\pi+9\pi}{36\pi}
14π36π\frac{14\pi}{36\pi}
718\frac{7}{18}

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