Answer:
Area of pathway = 10.375 unit²
Step-by-step explanation:
Given - Peter has built a gazebo, whose shape is a regular heptagon, with
a side length of 1 unit. He has also built a pathway around the
gazebo, of constant width 1 unit, as shown below.
To find - Find the area of the pathway.
Proof -
Central angle = [tex]\frac{360}{7}[/tex] = 51.43°
Now,
Central angle / 2 = [tex]\frac{360}{14}[/tex] = 25.71°
And
height = [tex]\frac{1}{2}.\frac{1}{tan25.41}[/tex] = 1.038
Now,
Inner area of gazebo = 7·[tex]\frac{1}{2}[/tex]·1·(1.038) = 3.634 unit²
Outer area of gazebo = ([tex]\frac{2.038}{1.038}[/tex])²·(3.364) = 14.009 unit²
∴ we get
Area of pathway = Outer area - Inner area
= 14.009 - 3.634
= 10.375 unit²
⇒Area of pathway = 10.375 unit²
