Answered By Leonardo F

First, let's review the original question:

Mia is designing a gazebo. Both the base and the roof will have the same regular hexagonal shape and size. the distance from the center of the base to a vertex on the base is 3m. 

A) What interior angle is formed at each vertex on the base?

B) The side length of the gazebos base is the same as a side length of the outer surface of the gazebos roof if the roof is 1.5m tall what angle is formed at the top of each triangular section of the roof?

Solution:

To answer letter A, we need to remember the following formula, which calculates the sum of the internal angles of any convex polygon:

 $Sum=180^o\left(n-2\right)$Sum=180^o(n−2) 

In which n is the number of sides in the polygon. Hence, in our case, for an hexagon, we have 6 sides:

 $Sum=180^o\left(6-2\right)=720^o$Sum=180^o(6−2)=720^o 

To calculate each individual angle in the hexagon, assuming it's regular (all sides and angles measure the same), each angle will be:

 $Angle=\frac{720^o}{6}=120^o$Angle=720^o6 =120^o 

B) The height of the roof forms the vertical leg of a right triangle, and the distance from the center to each vertex forms the horizontal leg. The hypotenuse is the slant of the roof. 

By definition, the tangent of the angle at the rooftop is opposite/adjacent: 

  $tan\left(\theta\right)=\frac{1.5m}{3m}=0.5\rightarrow\theta\approx27^o$tan(θ)=1.5m3m =0.5→θ≈27^o