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Some trigonometry determines that a circle inscribed in an equilateral triangle has a radius of 2/√3 times the side of the triangle. So the area of the circle is π(4/3). The “unused” corners are 1/3 the side of the original triangle.

So, if the area of the central circle is

A_0and the area of the smaller circles isA_1,A_2,A_3, …, then the total area we are looking for is:The sum of the geometric progression is:

Hence:

The circles fill approximately 0.831 times the area of the triangle (total area = 144√3).

Solution:The total area of all the circles is 66 π cm².Here’s a program that produces successive approximations of the sum: