Question

find the perimeter, the measure of a central angle, the length of an apothem, and the area for this regular polygon. round your answers to the nearest tenth.

Answer 1

Given:

Number of sides of the polygon = 5

Length of one side = 10 m

Let's solve for the following:

• (a). Perimeter

To find the perimeter of the polygon, we have:

P = n x L

Where:

n is the number if sides

L is the length of one side

Hence, we have:

P = 5 x 10 = 50m

The perimeter of the polygon is 50 m.

• (b). The measure of the central angle.

To find the measure of the central angle, we have:

`m=(360)/(n)=(360)/(5)=72^o`

Therefore, the measure of a central angle of the five sided polygon is 72 degrees.

In this question, angle given is the central angle divided by 2.

Thus, the indicated angle in the figure given in this question is:

`(72)/(2)=36^o`

The measure of the indicated angle is 36 degrees.

• Length of apothem:

Now, to find the length of the apothem, apply trigonometric ratio for tangent.

Thus, we have:

`tan\theta=(opposite)/(adjacent)`

Where:

opposite side is the side opposite the angle = 5 m

Adjacent side is the apothem.

θ is the angle = 36 degrees.

Hence, we have:

`\begin{gathered} tan36=(5)/(x) \\ \\ x=(5)/(tan36) \\ \\ x=6.9\text{ m} \end{gathered}`

Therefore, the length of the apothem is 6.9 m.

• Area of the polygon:

To find the area of the polygon, apply the formula:

`A=(1)/(2)*P*a`

Where:

P is the perimeter = 50 m

a is the length of the apothem = 6.9 m

Plug in the values and solve for the Area, A:

`\begin{gathered} A=(1)/(2)*50*6.9 \\ \\ A=172.5\text{ m}^2 \end{gathered}`

Therefore, the area of the polygon is 172.5 m².

ANSWER:

• Perimeter = 50 m

,

• Central angle = 72 degrees

,

• Length of apothem = 6.9 m

,

• Area of polygon = 172.5 m²