Answer:
389.1 units² (nearest tenth)
Step-by-step explanation:
Regular polygon: all side lengths are equal, all interior angles are equal.
Apothem: a line drawn from the center of any polygon to the midpoint of one of the sides
Radius: a line drawn from the center of the polygon to a vertex.
Therefore, we have been given the apothem of this regular dodecagon.
Formulae
[tex]\textsf{Area of a regular polygon}=\dfrac{n\:l\:a}{2}[/tex]
where:
- n = number of sides
- l = length of one side
- a = apothem (the line drawn from the center of any polygon to the midpoint of one of the sides)
[tex]\textsf{Length of apothem (a)}=\dfrac{l}{2 \tan\left(\frac{180^{\circ}}{n}\right)}[/tex]
where:
- l = length of one side
- n = number of sides
Solution
First, calculate the length of one side of the regular dodecagon by substituting a = 11 and n = 12 into the apothem formula:
[tex]\implies 11=\dfrac{l}{2 \tan\left(\frac{180^{\circ}}{12}\right)}[/tex]
[tex]\implies l=11 \cdot 2 \tan\left(\frac{180^{\circ}}{12}\right)[/tex]
[tex]\implies l=44-22\sqrt{3}[/tex]
Now substitute n = 12, the found value of l, and a = 11 into the area formula:
[tex]\implies \textsf{Area}=\dfrac{12(44-22\sqrt{3})(11)}{2}[/tex]
[tex]\implies \textsf{Area}=389.0622274...[/tex]
[tex]\implies \textsf{Area}=389.1\: \sf units^2 \: (nearest\:tenth)[/tex]
