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Sum of infinite geometric series

A regular hexagon is inscribed in a circle of radius 100. The area of that hexagon outside the square is shaded. Another circle is inscribed in the hexagon, and hexagon is inscribed in the second circle. The area of the second circle which lies outside the hexagon is shaded. This process continues to infinity.
What is the sum of all shaded areas?

Radius of 1 = 100
Radius of 2 = ______
Radius of 3 = _______
Radius of 4 = ________
Side of hexagon 1st =_________
Side of hexagon 2nd = ________
Side of hexagon 3rd = ________
Side of hexagon 4th = _______
Area of Circle 1 = ________
Area of Circle 2 = ________
Area of Crcle 3 =________
Area of Circle 4 = _______
Area of Hexagon 1 = _______
Area of Hexagon 2 =______
Area of Hexagon 3 = ______
Area of Hexagon 4 = _________
Difference of 1st =_________
Difference of 2nd =_________
Difference of 3rd =_________
Difference of 4th =________

Each new one is _________of the previous area.
Sum of all areas = ___________

Is there any relationship concerning the areas as the number of sides of the inscribed figure increases? If so what would you predict would be the area sum for 8 or 9 sides?_________

Solution Summary

This is an infinite geometric series question using the hexagons and circles inscribed inside of each other.

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