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# Geometric Series : Infinite Series of Circles inside Equilateral Triangles

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An equilateral triangle is inscribed in a circle of radius 100. The area of the circle which lies outside of the triangle is shaded. The process continues to infinity.

What is the radius for the second area/ third area/ fourth area?

Side of first area/ side of second area/ side of third area/ side of fourth area?

Area of first area is ________, area of second area is _________, area of third area is ______________, area of fourth area is __________

Each new area is ___________of the prevous area. What is the sum of all of the shaded areas?

##### Solution Summary

A recursive expression is found for the area outside a triangle but inside a circles for an infinite set of triangles and circles. The solution is detailed and well presented. Diagrams are included.

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Problem:

An equilateral triangle is inscribed in a circle of radius 100. The area of the circle which lies outside of the triangle is shaded. the process continues to infinity.
What is the radius for the second area/ third area/ fourth area?
side of first area/ side of second area/ side of third area/ side of fourth area?
area of first area is ________, area of second area is _________, area of third area is ______________, area of fourth area is __________
Each new area is ___________of the previous area.
What is the sum of all of the shaded areas?

Solution:

If (a1) is the side of the equilateral triangle inscribed in circle (R1), then
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