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21.4ft^2
Let's consider the given polygon.
We will first calculate the side length to find the perimeter. Then, we will find the apothem. Finally, since the area of a regular polygon is half the product of the apothem and the perimeter, we will use the obtained values to find the area. Let's do it!
By drawing the 5 radii we can divide the polygon into 5 isosceles triangles. Since the triangles are congruent and a full turn measures 360^(∘), the central angles of the isosceles triangles formed by the radii measure 3605= 72^(∘).
Recall that the apothem bisects the angle and the side of the regular polygon. Therefore, we obtain a right triangle with an acute angle that measures 36^(∘). Let x be the shorter leg of this right triangle.
Substitute values
LHS * 3=RHS* 3
Rearrange equation
Use a calculator
Round to 1 decimal place(s)
Once more, let's consider the right triangle whose longer leg is the apothem of the polygon.
Substitute values
LHS * 3=RHS* 3
Rearrange equation