Calculating radius of bigger circle If radius of  smaller congruent circles is equal to 20, what is radius of bigger circle?

 A: Assuming you have knowledge of basic trigonometry, you can proceed as follows:
Draw a triangle whose vertices are the centers of the $3$ smaller balls. Can you tell what is the length of the sides of that triangle? Can you tell what are its inner angles?
Sketch that triangle and the center $S$ of the bigger ball. What is its relative position to the triangle? Given that you know the inner angles and the sides of the triangle you should be able to find the distance between a vertex of the triangle and the center $S$ of the ball.
Now that you can tell the distance between $S$ and any of the ball's centers, can you tell what is the final radius of the bigger ball?
The final answer should be: (hover the yellow area)

 $20 + \frac{40}{\sqrt3}$

A: Let $z$ be the center of the circle in the bottom and let $o$ be the center of the big circle. Let $r$ be the radius of the small circle.
If we can find the length $l$ of segment $oz$ we are done, as the radius of the big circle is $r+l$.
In order to find $l$ notice that it is the center of gravity of the triangle formed by the centers of the small circles. This triangle is an equilateral triangle with side length $2r$. Since medians cut each other in ratio $1:2$ what we want is $\frac{2h}{3}$ where $h$ is the height of the triangle. It is clear that $h=\sqrt{3}r$.
We conclude $l=\frac{2}{\sqrt{3}}r$.
So the big radius is $r(1+\frac{2}{\sqrt{3}})$
