Area of a Regular Polygon 2

Area of a Regular polygon of n sides when the radius r of the inscribed circle is given:
Since \angle AOB = \frac{{{{360}^ \circ }}}{n}
\therefore  Area of the polygon  = n \times area of the \Delta AOB = n  \times \frac{{AB \times OD}}{2}


But OD = r, \frac{{AD}}{{OD}} =  \tan \frac{{{{180}^ \circ }}}{n}
\therefore  AD = OD\tan  \frac{{{{180}^ \circ }}}{n} and AD = 2r\tan  \frac{{{{180}^ \circ }}}{n}
Hence, area of the regular polygon  = n \times  \frac{{2r\tan \frac{{{{180}^ \circ }}}{n} \times r}}{2} = n{r^2}\tan  \frac{{{{180}^ \circ }}}{n}
Also, perimeter of the polygon  = nAB = n{\text{  }}2r\tan \frac{{{{180}^ \circ }}}{n} = 2nr\tan \frac{{{{180}^ \circ }}}{n}
(Putting for AB from above)

  \begin{gathered} A = n{r^2}\tan \frac{{{{180}^ \circ }}}{n} \\ P = 2{\text{ }}nr\tan \frac{{{{180}^ \circ  }}}{n} \\ \end{gathered}


A regular octagon circumscribed a circle of 2m radius. Find the area of the octagon.


Here n = 8, r = 2m

\therefore  Area of the polygon  = n{r^2}\tan  \frac{{{{180}^ \circ }}}{n} = 8 \times {(2)^2}\tan \frac{{{{180}^ \circ }}}{8}  = 8 \times 4 \times \tan {22.5^ \circ }

\therefore  Area of the polygon = 32 \times 0.4142  = 13.3 = 13.2544 = Square meter approx.