Question

I have tried to figure out this problem, but I keep on getting answers that aren’t on the slide. I looked it up on the community and they say that the answer is 663.1 square units, but they don’t explain how they got it. Can you help me?

Answer 1

Answer:

The area of the decagon = 663.1 units² (3rd option)

Step-by-step 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:

`\begin{gathered} Area\text{ of a polygon = }(1)/(2)* perimeter\text{ }*\text{ apothem} \\ Area\text{ = }(1)/(2)\text{Pa} \end{gathered}`

The perimeter of a polygon = length of one of the sides × number of sides

polygon is a decagon, number of sides = 10

`\begin{gathered} Perimeter\text{ = 9.3 }*\text{ 10} \\ Perimeter\text{ = 93 units} \end{gathered}`

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

`\begin{gathered} cos\text{ 18 = }(adjacent)/(hypotenuse) \\ cos\text{ 18 = }(a)/(15) \\ a\text{ = 15cos18} \\ a\text{ = 14.2658} \\ apothem\text{ = 14.27 \lparen approximately\rparen} \end{gathered}`

We can substitute the perimeter and apothem into the formula for the area of the polygon:

`\begin{gathered} Area\text{ of the decagon = }(1)/(2)*93*14.2658 \\ \\ Area\text{ of the decagon = 663.36 units}^2 \end{gathered}`

Due to approximation, the answer is slightly different from the workings

The area of the decagon = 663.1 units² (3rd option)