Question

The radius of a circle is 18. Find the length of the perimeter and the apothem of a regular (a) inscribed quadrilateral, and (b) circumscribed hexagon.

Answer 1

An apothem of a regular polygon will always be a radius of the inscribed circle. It is also the minimum distance between any side of the polygon and its center.

(a) Consider isosceles right triangle $N_1OA$

$$OA=AN_1=\dfrac{R}{\sqrt{2}}=\dfrac{18}{\sqrt{2}}=9\sqrt{2}$$

$$N_1N_4=2AN_1=18\sqrt{2}$$

$$Perimeter=4N_1N_4=72\sqrt{2}$$

$$apothem=OA=9\sqrt{2}$$

(b) The circumscribed hexagon is defined as the hexagon is outside in the circle or circle is inside in the hexagon.

Consider equilateral triangle $M_1OM_6$

$$\angle M_1OM_6=60\degree$$

Consider right triangle $M_1OB$

$$OB=R=18, \angle M_1OB=30\degree$$

$$OM_1=\dfrac{OB}{\cos\angle M_1OB}=\dfrac{18}{\sqrt{3}/2}=12\sqrt{3}$$

$$M_1M_6=OM_1=12\sqrt{3}$$

$$Perimeter=6M_1M_6=72\sqrt{3}$$

$$apothem=OB=18$$