A circle of unit radius is completely covered by four identical regular hexagons.

What is the smallest area of each hexagon?

Consider my previous solution.  Take the two hexagons that don't overlap, rotate them 30 degrees.  Then slide them towards each other until the two vertices of each touch sides of each of the other two hexagons.

Considering hexagons of unit side length:  the distance from the midpoint of the side the non-overlapping hexagons to the center of the rotated ones is (9-2√(3))/6.

The radius of the circle that fits is then √((43-12√(3))/12)≈1.306

If the circle has unit radius, the hexagon side length is the reciprocal of this: √(12/(43-12√(3)))≈.7350
And hexagon area (648+774√(3))/1417≈1.4034

Posted by Jer on 2016-07-07 14:13:26