Geometric Series : Infinite Series of Circles inside Equilateral Triangles
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?
© BrainMass Inc. brainmass.com March 6, 2023, 1:39 pm ad1c9bdddfhttps://brainmass.com/math/triangles/geometric-series-infinite-series-circles-inside-equilateral-triangles-40707
Solution Preview
Please see the attached file for the complete solution.
Thanks for using BrainMass.
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
...
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.