Hawaiian Earring

In terms of the radius of the smaller circle

Let the radius of the smaller circle be $r$, so the area of the smaller circle is $\pi r^2.$

The larger circle has radius $2r$ since its radius is equal to the diameter of the smaller circle, so the area of the larger circle is $\pi\left(2r\right)^2=4\pi r^2$.

The area of the part of the larger circle that is outside the smaller circle $4\pi r^2-\pi r^2=3\pi r^2.$

So the fraction of the larger circle which is outside the smaller circle is $\dfrac{3\pi r^2}{4\pi r^2}=\dfrac{3}{4}.$


In terms of the radius of the larger circle

Let the radius of the larger circle be $r$, so the area of the larger circle is $\pi r^2.$

The smaller circle has radius $\frac{r}{2}$ since it has diameter $r$, so the area of the smaller circle is $\pi\left(\frac{r}{2}\right)^2=\dfrac{\pi r^2}{4}$.

The area of the part of the larger circle that is outside the smaller circle $$\pi r^2-\dfrac{\pi r^2}{4}= \pi r^2 \left(1-\frac{1}{4}\right)=\pi r^2\times\frac{3}{4}$$
Which is $\frac{3}{4}$ of $\pi r^2$, the area of the larger circle.


Using area scale factors

The diameter of the smaller circle is equal to the radius of the larger circle, so the length scale factor between the larger circle and the smaller circle is $\frac{1}{2}.$

That means that the area scale factor between the larger circle and the smaller circle is $\left(\frac{1}{2}\right)^2=\frac{1}{4}$, which means that the area of the smaller circle is $\frac{1}{4}$ of the area of the larger circle.

So the area of the rest of the larger circle is $\frac{3}{4}$ of the area of the larger circle.