Maximum area of circles inscribed in sectors of the unit circle

The ratio of the total area of the disks over the outer circle is the same as the ratio of the area of a single disk over the tight annular sector that bounds it.

If there are $k$ circles of radius $r$ in the outer ring, the subtended angle is $\theta=\dfrac{2\pi}k$.

The area of a sector is $$\theta(1-(1-2r)^2)$$

and the requested ratio

$$\frac{kr}{4(1+2r)}.$$