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 .

The answer is 120.

1 solution

Brian Kardon
Feb 24, 2015

We can express the fraction $f$ of the poster that is black in terms of the known quantities and $f$ itself:

$f = \frac{f A_c + A_{outer} - A_{inner}}{A_{outer}}$

Where $A_c$ is the total area of the pie chart, $A_{outer}$ is the area of the entire poster, and $A_{inner}$ is the area of the white internal rectangle inside the border.

Solving this for $f$ , we get

$f = \frac{A_{outer} - A_{inner}}{A_{outer} - A_c}$

Plugging in the values

$A_{outer} = 200~in^2$

$A_{inner} = 144~in^2$

$A_c = \pi (4 \sqrt{\frac{2}{\pi}})^2 = 32~in^2$

We find that $f = \frac 1 3$ . If a third of the circle should be black, that translates to an angle of 120°.

Self-referential pie chart

The answer is 120.

1 solution

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