This is the old "Kissing Circles" problem for four mutually tangent circles. I didn't immediately see the geomtry-based solution, but I remembered there was an algebra formula (and a poem!) that gives the radius X of the larger circle if you know the radii A, B, and C of the smaller circles:
X = ABC/(2*sqrt(ABC(A+B+C)) - AB - AC - AB)
Using A = B = C = 20, you get X = 20 + 40/sqrt(3). The poem is called "The Kiss Precise"
You can see an interactive applet of the circle problem here
and the attendant math stuff here
Since Larry was kind enough to answer my question about hurricanes, I hope I can try to clarify why knowing the sizes of the three circles will give you enough information to find the size of the larger circle.
If you're given this type of unknown radius problem, then in order to solve it, you need *some* other information that will allow you to work backwards to solve for the radius. The simplest example of such a problem would be "A circle's diameter is 6, what is the radius?" In this example, you divide the diameter by 2 to get r = 6/2 = 3. There is a precise relation between diameter and radius that allows you to find the radius if you know the diameter.
A slightly more complicated example would be "A circle's area is 9*pi, what's the radius?" For this you need the formula Area = pi*r*r. Then you solve 9*pi = pi*r*r to get r = 3. There is a precise relation between area and radius that allows you to find the radius if you know the area.
For this 3-circle problem, there is a precise relation between the radius of a small circle and the radius of the large circle. If you put three equally sized circles together like this, there is only ONE way to fit a larger circle around them so that they all fit together snuggly. Any other circle you construct would be too small or too big. Once you figure out that precise relation, you can use information about a small circle's radius to calculate the radius of the larger circle.
It's not that the three circles are necessary, it's that you need some information that bears a unique relationship to the radius. In this problem, the three mutually tangent circles provide that. I hope I explained myself well.