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    In the figure (see attachment) there are infinitely many circles approaching the vertices of an equilateral triangle, each circle touching other circles and sides of the triangle. If the triangle has sides of length 1, find the total area occupied by the circles.

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    Solution Preview

    All the circles are inscribed in some equilateral triangles. So we need to know the following things:

    1. Given an equilateral triangle, what is the area of the inscribed circle
    2. What is the side length of three little equilateral triangles in the three corners (see attachment)

    1. Let the side length of an equilateral triangle be L.

    a) Find the height H (see Fig.2) and the area A of the triangle: from the Pythagoras Theorem

    Solution Summary

    This shows how to find the area of infintely many circles in a triangle.