Area

Find the area of a regular hexagon with apothem length of 5 centimeters

a = apothem

a = 5 cm

$\begin{array}{rcll} \mathbf{ A_n } & \mathbf{=} & \mathbf{ n\cdot a^2 \cdot \tan( \frac{180^{\circ}}{n} ) } \end{array}$

hexagon: n = 6

$\begin{array}{rcll} A_6 &=& 6\cdot a^2 \cdot \tan( \frac{180^{\circ}}{6} )\\ A_6 &=& 6\cdot 5^2 \cdot \tan( 30^{\circ} ) \qquad &| \qquad \tan( 30^{\circ} )=\frac13 \cdot \sqrt{3} \\ A_6 &=& 6\cdot 5^2 \cdot \frac13 \cdot \sqrt{3} \\ A_6 &=& 2\cdot 5^2 \cdot \sqrt{3} \\ A_6 &=& 2\cdot 25 \cdot \sqrt{3} \\ A_6 &=& 50 \cdot \sqrt{3} \\ A_6 &=& 86.6025403784\ \text{cm}^2 \\ \end{array}$

The area of a regular hexagon with apothem length of 5 centimeters is $86.6\ \text{cm}^2$

see:http://www.mathwords.com/a/area_regular_polygon.htm