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 the area of the polygon  = n \times area of the \Delta AOB = n \times \frac{{AB \times OD}}{2}


polygon-02

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, the 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, the perimeter of the polygon  = nAB = n{\text{ }}2r\tan \frac{{{{180}^ \circ }}}{n} = 2nr\tan \frac{{{{180}^ \circ }}}{n}
(Using 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}

 

Example:

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

 

Solution:

Here n = 8, r = 2m

\therefore the 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 the area of the polygon = 32 \times 0.4142 = 13.3 = 13.2544 = square meters, approximately.