Start by evaluating the side length using the perimeter.
Yes, the area is approximately 24.75 square inches. For explanation, see solution.
Practice makes perfect
We are given that the perimeter of a regular 9-gon is 18 inches, and we are asked to determine whether this is enough information to evaluate the area of this figure. First let's recall the formula for the area of a regular polygon.
A=1/2ans
In this formula a is the apothem, n is the number of sides, and s is the side length. The number of sides in a regular nonagon is 9. Knowing the perimeter we are able to find s, as in a regular polygon all sides have the same length.
s=18/9=2
Each side has a length of 2 inches. Let's draw our polygon and think about how to evaluate the apothem a, keeping in mind that a regular 9-gon can be divided into nine congruent isosceles triangles.
Next let's find the central angle of this polygon. To do this we need to divide 360^(∘) by the number of sides, 9.
360^(∘)/9=40^(∘)
Now, notice that apothem bisects the central angle as well as the side.
By using the tangent ratio we can write an equation for a.
The apothem of this polygon is approximately 2.75 inches. As we can see, we found all the information we need to evaluate the area. Let's substitute them into the formula.
The area of this polygon is approximately 24.75 square inches. Notice that this value is only an approximation, as we used approximate value of the apothem to evaluate it.
