A Different Sort of Pie Chart

Fractions are often represented as pie charts, where a circle is broken into slices called sectors.

We can use the relationship between fractions and circle sectors to calculate the area of the sectors. For example, the central angle of the dark-shaded sector above is one-eighth of the circle, so its area is $$\frac18$$ the area of the circle.

But what happens if—instead of cutting a circle up—we stretch it out and multiply its area?

It turns out that we can think of an ellipse as a circle that has been stretched or compressed vertically and/or horizontally. The unit circle below has a radius of 1. Changing the value of $$a$$ from 1 scales the circle horizontally, since $$a$$ is the distance from the center of the ellipse to either of its points on the $$x$$-axis. Similarly, $$b$$ is the distance from the center of the ellipse to either $$y$$-intercept of the ellipse; changing its value scales the ellipse vertically.

So what happens to the area of the unit circle when we distort it this way? Its original area is $$A=\pi r^2 = \pi(1)^2 = \pi,$$ while setting $$a=3$$ to stretch the circle horizontally by a factor of 3 will multiply the area it covers by 3. Similarly, setting $$b=\frac{1}{2}$$ to squeeze the circle vertically so it's only $$\frac{1}{2}$$ as tall will reduce its area by a factor of $$\frac{1}{2}.$$ Doing both makes the new area $$A=\pi (3)\left(\frac{1}{2}\right)=\frac{3\pi}{2};$$ this ellipse has $$\frac{3}{2}$$ the area of the unit circle. In general, we can find the area of any ellipse—including a circle!—as $$A=\pi ab.$$

Today's Challenge

The diameters of the circles below get cut in half as the circles get smaller.

What fraction of the largest circle is contained inside of the orange region?

Note: The area of an ellipse can be found by $$A=\pi ab,$$ where $$a$$ and $$b$$ are the longest and shortest distances from a point on the ellipse to its center.

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