Find the area of each regular polygon below using the given information. A regular pentagon with a side of 8. (Round answer to the nearest tenth.)
The answer is 110.1 units².
Draw an apothem from the center to the pentagon to one of the sides. Next, draw a radius, forming a triangle within the pentagon. As one interior angle of a regular pentagon is 108 degrees, and the radius bisects it, the measure of the angle enclosed in the triangle is 54 degrees. Since the apothem is perpendicular to the side, we know that the angle at which they intersect is a 90 degree, or right angle. You can optionally find out that our third triangle angle measure is 36 because the sum of the angles of a triangle is 180.
Now, using our trigonometric ratios (tan, cos, sin) we can determine the length of the apothem. We can write that the (side adjacent to 36)/(side opposite to 36)=tan 36. The side adjacent to 36 is 4 (half of 8; the apothem bisects the side) and the side opposite to 36 is the apothem, the length of which is unknown.
We will now flip both of our fractions so that the new fractions are (apothem)/(4)=1/(tan 36). The tan of 36 is approximately .7265. Upon solving 1/.7265 and multiplying it by 4, we get 5.5058499… This is the length of the apothem.
As the formula for the area of a polygon is 1/2(apothem)(perimeter), and the perimeter is 40 (8*5) and the apothem is 5.5058499, the area is 110.116999…
Rounded to the nearest tenth, you will get the above answer. I hope that wasn’t too confusing, though without writing it down, it is quite hard to see what I mean.