What is the area of a regular nonagon with an apothem of 4 units?

To find the area of a regular nonagon (9-sided polygon) with an apothem of 4 units, you can use the formula for the area of a regular polygon:

Area = 1/2 * Perimeter * Apothem

First, you need to find the perimeter. The perimeter of a regular nonagon is 9 times the length of one side. However, we don't have the side length. But we can find it using the relationship between the side length (s), the apothem (a), and the number of sides (n) in a regular polygon, which is given by the formula:

s = 2a * tan(π/n)

Substituting the given values, we get:

s = 2 * 4 * tan(π/9) = 8 * tan(π/9)

Now, we can find the perimeter (P) by multiplying the side length by the number of sides:

P = 9 * s = 9 * 8 * tan(π/9)

Finally, we can find the area (A) by substituting the values of the perimeter and the apothem into the area formula:

A = 1/2 * P * a = 1/2 * 9 * 8 * tan(π/9) * 4 = 18 * 4 * tan(π/9)

So, the area of the regular nonagon with an apothem of 4 units is 72 * tan(π/9) square units.