An equilateral triangle with unit side is completely covered by six identical circles.
What is the smallest radius of each of the circles?
Divide the triangle into 9 smaller triangles with side length 1/3. There will be six small triangles in the same orientation as the main triangle and the other three will be in the opposite orientation.
Draw the circumcircles of the group of six small triangles. The other three small triangles are covered by the arcs of these circles. Then these six circles cover the entire main triangle.
Each of the circles has a radius of 1/sqrt(27) = 0.19245