# Solution to the Two Regular Hexagons (III) Puzzle +-- {.image} [[TwoRegularHexagonsIII.png:pic]] > Both hexagons are regular. What’s the angle? =-- ## Solution by Calculating Angles To calculate the indicated angle, consider the rest of the turn. This consists of two interior angles of hexagons with an overlap. The [[interior angles in a polygon|interior angle]] of a [[hexagon]] is $120^\circ$. There are a couple of ways to calculate the overlap. The green area is an [[isosceles triangle]], with apex angle the interior angle of the smaller hexagon, so the base angle (which is the overlap angle) is half of $180^\circ - 120^\circ$, which is $30^\circ$. Alternatively, the base of that line joins two parallel sides of the hexagon and so by symmetry is perpendicular to those sides. The overlap angle is therefore $120^\circ - 90^\circ = 30^\circ$. The angle to be calculated is therefore: $$ 360^\circ - (240^\circ - 30^\circ) = 150^\circ $$