What fraction is shaded? Conrguent $\frac{5}{6}$ circles in a circle. What fraction is shaded?
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Solution:

Let $r$ be the radius of the small circles and $R$ the radius of the big one.
The colored section is three times five sixths of the area of one of the small circles.
Colored Section area= $3\times\dfrac{5}{6}\times\pi r^2=\dfrac{5\pi r^2}{2}$
The radius $R$ of the big circle is equal to $r$ plus the radius of the circumscribed circle of equailateral triangle ABC, whose side is $2r$.
The radius of the circumscribed circle of an equilateral triangle is the length of the sides divided by $\sqrt{3}$. Since the side here measures $2r$, the radius of the circumscribed circle is $\dfrac{2r}{\sqrt{3}}$.
So we have  $R = r+\dfrac{2r}{\sqrt{3}}$
The area of the big circle is $\pi \times R^2$, which here is equal to $(r+\dfrac{2r}{\sqrt{3}})^2$
which, when expanded, gives
Big Circle area = $\dfrac{\pi r^2(7+4\sqrt{3})}{3}$
To obtain the shaded fraction, we need to divide the area of the colored region by the area of the big circle:
Shaded fraction = $\dfrac{\dfrac{5\pi r^2}{2}}{\dfrac{\pi r^2(7+4\sqrt{3})}{3}}$
Shaded fraction = $\dfrac{5\pi r^2}{2} \times \dfrac{3}{\pi r^2(7+4\sqrt{3})}$
Shaded fraction = $\dfrac{15}{2(7+4\sqrt{3})} \simeq 53.847 \% $
I think it's wrong. In the drawing the smaller circles are not tangent to the largest

We need to find the radius of the large circle and the rest is straight forward:

