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### Best Answer Jim Bob says

The pink colored in triangle I drew has angles of 30, 60, and 90 degrees. That means 20 equals x*sqrt(3)/2, or x = 40/sqrt(3). Since x + 20 is the radius of the larger cirlce, the answer is exactly 20 + 40/sqrt(3), or about 43.094.

Good work there.

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### I'M BANNED Y'ALL!!!!!!!!!! says

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"

http://www.pballew.net/soddy.html

You can see an interactive applet of the circle problem here

http://www.had2know.com/academics/kissing-circles-...

and the attendant math stuff here

http://en.wikipedia.org/wiki/Descartes%27_theorem

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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.

Descartes' Theorem. That's a much faster way to solve it for sure.

### Larry Wall says

A circle is 360 degree. Its diameter is 180 degrees. Will assign the diameter a value of X to represent its length. Therefore, regardless of what objects are inside the circle and what their radius may be has no bearing on the actual size of the circle. Since the radius of a circle is one half of the diameter, we can conclude that the radius of the circle is the sum of R = x/2. Since the diameter has to equal 180 degrees, the radius must be 90 degrees. ..

I stand to be corrected. Math and geometry were not my subjects. If you are looking for a numerical number, then I have no way of figuring it out,

I accept the answer, I just wish you would tell me where I was wrong in my approach. I still contend that the smaller circles had no impact on the radius of the larger circle. Would the radius be different if the smaller circles were not there?

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### normal says

If we join all the center point of the small circles then it will form a equilateral triangle, then by applying the formula for midpoint of the equilateral triangle and then

Length of midpoint of triangle to one of the vertex + Small circle radius(ie. 20) = radius of the larger circle

You're on the right track. You do need to examine what's going on in the center of the circle and at the points of tangency.