Let's start by using the fact that the sum of the exterior angles of any polygon is always 360 degrees. Since we know that the measure of one exterior angle of this regular polygon is 18 degrees, we can find the number of sides of the polygon as follows:
360 / 18 = 20
So the polygon has 20 sides. Now, to find the area of the polygon, we need to know the length of one of its sides and its apothem (the distance from the center of the polygon to the midpoint of a side).
Since the polygon is regular, all of its sides are congruent and all of its apothems are congruent as well. We can use trigonometry to find the length of the apothem, using the fact that the tangent of half of one of the interior angles of the polygon is equal to the ratio of the apothem to half of a side length.
The measure of each interior angle of the polygon can be found as follows:
180 - 18 = 162
Since the polygon is regular, all of its interior angles are congruent. We can use the fact that the sum of the interior angles of an n-sided polygon is (n-2)*180 degrees to find the measure of one interior angle:
(n-2)*180 / n = 162
Solving for n, we get:
n = 10
So the polygon has 10 interior angles, and each interior angle measures 162 degrees. Half of an interior angle measures 81 degrees, so we can use the tangent of 81 degrees to find the apothem:
tan(81) = apothem / (side length / 2)
Solving for the apothem, we get:
apothem = (side length / 2) * tan(81)
Now we just need to find the side length. We can use the fact that the apothem and the side length form a right triangle with one of the interior angles of the polygon. The sine of half of one of these interior angles is equal to the ratio of the apothem to the hypotenuse of this right triangle (which is half of a side length).
The measure of one of these interior angles can be found as follows:
180 - 162 = 18
So half of one of these interior angles measures 9 degrees. Using the sine of 9 degrees, we can find the side length:
sin(9) = apothem / (side length / 2)
Solving for the side length, we get:
side length = 2 * apothem / sin(9)
Now we can find the area of the polygon using the formula:
area = (1/2) * perimeter * apothem
where perimeter is the total length of all of the sides of the polygon. Since the polygon is regular, the perimeter is just the product of the number of sides and the length of one side:
perimeter = 10 * side length
Putting it all together, we get:
apothem = (side length / 2) * tan(81)
side length = 2 * apothem / sin(9)
perimeter = 10 * side length
area = (1/2) * perimeter * apothem
Plugging in the values, we get:
apothem = (side length / 2) * tan(81) = (side length / 2) * 6.3138 = 0.8333 * side length
side length = 2 * apothem / sin(9) = 28.2791
perimeter = 10 * side length = 282.791
area = (1/2) * perimeter * apothem
= (1/2) * 282.791 * 0.8333 * side length
= 117.462 square units