Answer:
The area of the decagon = 663.1 units² (3rd option)
Explanation:
Given:
The polygon is a decagon
radius = 15 units
length of one of the side of the polygon = 9.3 units
To find:
The approximate area of the polygon
The formula for area of a polygon:
[tex]\begin{gathered} Area\text{ of a polygon = }\frac{1}{2}\times perimeter\text{ }\times\text{ apothem} \\ Area\text{ = }\frac{1}{2}\text{Pa} \end{gathered}[/tex]
The perimeter of a polygon = length of one of the sides × number of sides
polygon is a decagon, number of sides = 10
[tex]\begin{gathered} Perimeter\text{ = 9.3 }\times\text{ 10} \\ Perimeter\text{ = 93 units} \end{gathered}[/tex]
To determine the apothem of the polygon, we need to do an illustration of the polygon:
Sum of angles at a point = 360°
Each angle in the triangle = 360/number of sides
Each angle = 360/10 = 36°
The apothem is the perpendicular height from the center of the polygon to one of the side of the polygon
We will apply cosine ratio CAH) to get the apothem
angle = 18°
opposite = side opposite the angle = not given
adjacent = a = apothem
hypotenuse = r = 15
[tex]\begin{gathered} cos\text{ 18 = }\frac{adjacent}{hypotenuse} \\ cos\text{ 18 = }\frac{a}{15} \\ a\text{ = 15cos18} \\ a\text{ = 14.2658} \\ apothem\text{ = 14.27 \lparen approximately\rparen} \end{gathered}[/tex]
We can substitute the perimeter and apothem into the formula for the area of the polygon:
[tex]\begin{gathered} Area\text{ of the decagon = }\frac{1}{2}\times93\times14.2658 \\ \\ Area\text{ of the decagon = 663.36 units}^2 \end{gathered}[/tex]
Due to approximation, the answer is slightly different from the workings
The area of the decagon = 663.1 units² (3rd option)
