Question #158207

The radius of a circle is 18. Find the length of the peri

meter and the apothem of a

regular

(a) inscribed quadrilateral,

and

(b) circumscribed hexagon.



1
Expert's answer
2021-01-29T12:11:24-0500

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 N1OAN_1OA


OA=AN1=R2=182=92OA=AN_1=\dfrac{R}{\sqrt{2}}=\dfrac{18}{\sqrt{2}}=9\sqrt{2}

N1N4=2AN1=182N_1N_4=2AN_1=18\sqrt{2}

Perimeter=4N1N4=722Perimeter=4N_1N_4=72\sqrt{2}

apothem=OA=92apothem=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 M1OM6M_1OM_6


M1OM6=60°\angle M_1OM_6=60\degree

Consider right triangle M1OBM_1OB


OB=R=18,M1OB=30°OB=R=18, \angle M_1OB=30\degree

OM1=OBcosM1OB=183/2=123OM_1=\dfrac{OB}{\cos\angle M_1OB}=\dfrac{18}{\sqrt{3}/2}=12\sqrt{3}

M1M6=OM1=123M_1M_6=OM_1=12\sqrt{3}

Perimeter=6M1M6=723Perimeter=6M_1M_6=72\sqrt{3}

apothem=OB=18apothem=OB=18




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