1020.44 – Dividing the Yin and the Yang

The familiar yin-yang symbol is the interior of a circle divided into two regions by the joining of two half-circles of half the radius of the big circle, as in the figure below. You are to draw a single straight line segment that divides each of the two areas, yin and yang, in half. Explain why your line segment works!

Solution

If we let the large circle have radius $2$ then each quadrant has area $\pi$ and each half of the small circle has area $\dfrac{\pi}{2}$ . This information is included the figure below, part [A].

It follows that yin and yang are each area $2 \pi$ and a 45° line will divide the figure the way we want. In the figure, part [B], on each side of the line there is $\pi$ worth of yin and ditto of yang.