Question

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. (Every point on the ground that is within $1$ unit of the gazebo and outside the gazebo is covered by the pathway.) Find the area of the pathway.

Answer 1

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 = $\frac{360}{7}$ = 51.43°

Now,

Central angle / 2 = $\frac{360}{14}$ = 25.71°

And

height = $\frac{1}{2}.\frac{1}{tan25.41}$ = 1.038

Now,

Inner area of gazebo = 7·$\frac{1}{2}$·1·(1.038) = 3.634 unit²

Outer area of gazebo = ($\frac{2.038}{1.038}$)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