Notes
two rectangles overlapping a circle solution

Solution to the Two Rectangles Overlapping a Circle Puzzle

Solution by Area of a Circle, Area of a Rectangle, and Lengths in a Regular Hexagon

As this is a comparison of areas, let us set the radius equal to $1$ unit. Then the red area is the area of the cross (formed from two overlapping rectangles) without the white region, and the yellow area is the area of the circle without the white region. Therefore, to compare the red and yellow areas it is sufficient to compare the area of the circle with the area of the cross.

As the dots are evenly spaced, and there are twelve of them, joining every other dot forms a regular hexagon. One segment of this, as shown, is an equilateral triangle meaning that the height of the rectangle is equal to the radius of the circle. This means that the dimensions of the rectangle are $2 \times 1$, and the central square has dimensions $1 \times 1$. So the area of the cross is $3$.

Since the area of the circle is $\pi$ and $\pi \gt 3$, the yellow area is larger than the red area.

For completeness, to work out the white area then split the white part of one rectangle into two sectors and two triangles. Each sector has area $\frac{\pi}{6}$. Each triangle has base $\sqrt{3}$ and height $\frac{1}{2}$ so has area $\frac{\sqrt{3}}{4}$. The white area inside one rectangle has area $\frac{\pi}{3} + \frac{\sqrt{3}}{2}$ so the area of the full white region is:

Therefore the red area is:

and the yellow area is:

The difference between these is $\pi - 3$ as before.