Question

What is the radius of the incircle of a regular hexagon with a side of 20 cm ?

Answer 1

10√3 cm.
The radius of an incircle of a hexagon is the same measure as the apothem. find the apothem by taking half of the central angle and create a right triangle perpendicular to the side. The central angle is 60 so the angle opposite the side length will be 30.  this creates a 30-60-90 triangle with the ratio x:x√3:2x. half the side opposite 30 is 10, so x=10. the base angle of the side is 60 so it will be x√3. plug in 10 for x to get 10√3 cm.