Self-referential pie chart

Geometry Level pending

As shown in the image, there is a 10"x20" white poster with a 1 inch black border on all sides. A black and white pie chart is in the center of the poster. The radius of the pie chart is \(4 \sqrt{\frac{2}{\pi}}\) inches.

If the black portion of the pie chart is to accurately represent the fraction of the poster's area that is black, what angle (in degrees) should the black part of the pie chart have?

You should assume that the thin black circular outline of the pie chart contributes negligible area.

Inspired by this xkcd cartoon.


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