Answer:
[tex] \displaystyle E)\: 162 \sqrt{3} [/tex]
Step-by-step explanation:
we are given a side a polygon
and said to figure out the area
recall the formula of regular polygon
[tex] \displaystyle \: \frac{ {na}^{2} }{4} \cot \left( \frac{ {180}^{ \circ} }{n} \right) [/tex]
where a represents the length of a side
and n represents the number of sides
the given shape has 6 sides
and has a length of [tex]\displaystyle 6\sqrt{3}[/tex]
so our n is 6 and a is 6√3
substitute the value of n and a:
[tex] \displaystyle \: \frac{ {6 \cdot \:( 6 \sqrt{3} })^{2} }{4} \cot \left( \frac{ {180}^{ \circ} }{6} \right) [/tex]
reduce fraction:
[tex] \displaystyle \: \frac{ {6 \cdot \:( 6 \sqrt{3} })^{2} }{4} \cot \left( \frac{ { \cancel{180}^{ \circ}} ^{ {30}^{ \circ} } }{ \cancel{6 \: } } \right) [/tex]
[tex] \displaystyle \: \frac{6 \cdot \: (6 \sqrt{ {3} } {)}^{2} }{4} \cot( {30}^{ \circ} ) [/tex]
simplify square:
[tex] \displaystyle \: \frac{6 \cdot \: 36 \cdot \: 3 }{4} \cot( {30}^{ \circ} ) [/tex]
reduce fraction:
[tex] \displaystyle \: \frac{6 \cdot \: \cancel{36} \: ^{9} \cdot \: 3 }{ \cancel{ 4 \: } } \cot( {30}^{ \circ} ) [/tex]
[tex] \displaystyle \: 6 \cdot \: 9 \cdot \: 3 \cot( {30}^{ \circ} ) [/tex]
simplify multiplication:
[tex] \displaystyle \: 162\cot( {30}^{ \circ} ) [/tex]
recall unit circle:
[tex] \displaystyle \: 162 \sqrt{3} [/tex]
hence, our answer is E
