A circle is drawn inside a regular hexagon of side 14 cm so that it touches all six sides. Find the area of that circle in ${\text{cm}}^2$. Take π = 22/7.

The radius of this circle is the distance from the centre to the midpoint of a side. Each of the six equilateral triangles has side 14 cm, so that distance is its height, (√3/2) × 14 = 7√3 cm, and $r^2=147$. Area = $(22/7)\times 147=22\times 21=462{\text{cm}}^2$. (Using the corner distance 14 cm instead would wrongly give $616{\text{cm}}^2.$)