To answer this question, we will use the following diagram as reference:
From the above diagram, we want to determine the length of HI and L. Now, we know that:
Since triangle HIB is a right triangle and α=360/7 degrees, then:
![\begin{gathered} \sin (\alpha)/(2)=((L)/(2))/(HB)\text{.} \\ \sin ((360)/(14))=\frac{L}{18\operatorname{mm}}\text{.} \end{gathered}]()
Therefore:
![L=\sin ((360)/(14)^(\circ))18\operatorname{mm}.]()
To determine HI we use the cosine function:
![\begin{gathered} \cos ((\alpha)/(2))=(HI)/(HB)=\frac{HI}{9\operatorname{mm}}, \\ HI=\cos ((360)/(14)^(\circ))9\operatorname{mm}. \end{gathered}]()
Finally, recall that the area of a heptagon is given by the following formula
Substituting the values we found we get:
![A=\frac{\cos ((360)/(14)^(\circ))9\operatorname{mm}(\sin ((360)/(14)^(\circ))18\operatorname{mm}*7)}{2}\approx221.649mm^2.]()
Answer:
