The side of a pentagon is 20 ft the inside has a right triangle that has a height of 13.8 and a side of 17 ft
Find the area of the regular pentagon shown below.
359 ft^2
286.7 ft^2
690 ft^2
882.2 ft^2
To find the area of the regular pentagon, we need to find the apothem first.
Since the right triangle inside the pentagon has a height of 13.8 and a side of 17 ft, we can find the base of the right triangle using the Pythagorean theorem:
Base of the right triangle = sqrt(17^2 - 13.8^2)
Base = sqrt(289 - 190.44)
Base = sqrt(98.56)
Base = 9.93 ft
The base of the right triangle is half the side of the pentagon, so the side of the pentagon is twice the base:
Side of the pentagon = 2 * 9.93
Side of the pentagon = 19.86 ft
Now, since we have the side of the pentagon, we can find the apothem using the formula:
Apothem = side / (2tan(180/n)) where n is the number of sides of the polygon
For a regular pentagon, n = 5, so:
Apothem = 19.86 / (2tan(180/5))
Apothem = 19.86 / (2tan(36))
Apothem = 19.86 / (2 * 0.7265)
Apothem = 13.7 ft
Now, we can find the area of the pentagon using the formula:
Area = (Perimeter * Apothem) / 2
Area = (5 * 20 * 13.7) / 2
Area = 685 ft^2
Therefore, the closest option is 690 ft^2.