Expert: Scott A Wilson - 12/23/2009

Question
Hi thanks so much for your time.  I have a problem that we can do for math for extra credit and i was wondering if you could help me with it.  My gymnastics coach majored in math and even she had some trouble with it. But Here it is:

SeeMore's Symbol: The official symbol of the society of Mathematical enthusiasts at seemore university is a circle with an equilateral triangle inscribed and another equaliteral triangle circumscribed about the circle.  The difference between the areas of the two triangles is 25cm squared. What is the radius of the circle? My teacher said there needs to be an approximate answer and an exact answer. If you could give me any help with this that would be great! There should be a picture attached if not it is an equilaterial triangle, then a circle inside that, then another equilater triangle inside the circle. Thank you so much.  

Answer
If there is a circle with a triangle on the outside and on the inside,
then the inner triangle is of 1/4 the area of the outer tringle.

Lets say the inside triangle has area X.  This means the outside triangle has area 4X.
This means that the difference is 3X, so solve the equation 3X = 25, so X=25/3.

Since the area is 4 times as big on the outside triangle,
this means that the sides are √4 times as long, and √4 = 2.

To answer the question on the radius of the circle,
the distance from a corner to the center of a triangle is √3/3
and the distace to the far side is √3/6.

What we are concerned about, though, is how far it is from the center of the triangle
to one of its corners, for that is the radius of the circle.  That is, √3/3.