# Solution to the Puzzle +-- {.image} [[TwoHexagonsandaTriangle.png:pic]] > All three polygons here are regular. The area of the small hexagon is $5$. What’s the area of the large one? =-- ## Solution by [[Angle at the Centre is Twice the Angle at the Circumference]] +-- {.image} [[TwoHexagonsandaTriangleLabelled.png:pic]] =-- With the points labelled as above, consider the circle centre $O$ through $A$. As the purple hexagon is [[regular]], $O B = O A$ so this circle also passes through $B$. The angle $A \hat{O} B$ is $120^\circ$ as it is the [[interior angle]] in a regular [[hexagon]]. Angle $A \hat{C} B$ is $60^\circ$ as the interior angle in an [[equilateral triangle]] which is half $120^\circ$. Hence by the fact that the [[angle at the centre is twice the angle at the circumference]], $C$ lies on the circle. Hence $O C = O A$. Now $O C$ is $\sqrt{3}$ times the side length of the smaller hexagon (see the page on [[hexagons]] for how the various lengths are related) so the scale factor from the red hexagon to the purple hexagon is $\sqrt{3}$ and hence the area scale factor is $3$. Thus the area of the larger hexagon is $3 \times 5 = 15$.