Exercise 23 Page 647

Let's consider the given polygon.

We will first calculate the side length to find the perimeter. Then, we will find the apothem. Finally, since the area of a regular polygon is half the product of the apothem and the perimeter, we will use the obtained values to find the area. Let's do it!

Side Length and Perimeter

By drawing the 3 radii we can divide the polygon into 3 isosceles triangles. Since the triangles are congruent and a full turn measures 360^(∘), the central angles of the isosceles triangles formed by the radii measure 3603= 120^(∘).

Recall that the apothem bisects the angle and the side of the regular polygon. Therefore, we obtain a right triangle with an acute angle that measures 60^(∘). Let x be the shorter leg of this right triangle.

This is a right triangle with a hypotenuse that measures 4m. To find the length of the side that is opposite to the acute angle that measures 60^(∘), we will use the sine ratio. sin θ =Opposite/Hypotenuse Let's substitute the corresponding values into this ratio and solve for x.

sin θ =Opposite/Hypotenuse

SubstituteValues

Substitute values

sin 60^(∘)=x/4

MultEqn

LHS * 4=RHS* 4

4sin 60^(∘)=x

RearrangeEqn

Rearrange equation

x=4sin 60^(∘)

The length of the shorter leg of the right triangle is 4sin 60^(∘) m. As previously mentioned, the apothem bisects the side of the polygon. Therefore, the side length is twice the length of the shorter leg of the triangle.

Finally, since we have a 3-sided regular polygon we can find the perimeter by multiplying 3 by the side length.

Perimeter= 3* 8sin 60^(∘)

UseCalc

Use a calculator

Perimeter= 20.784609...

RoundDec

Round to 1 decimal place(s)

Perimeter ≈ 20.8

The perimeter of the given regular polygon is 20.8m, to the nearest tenth.

Apothem and Area

Once more, let's consider the right triangle whose longer leg is the apothem of the polygon.

Remember, this a right triangle with a hypotenuse that measures 4m. To find the length of the side that is adjacent to the acute angle that measures 60^(∘), we will use the cosine ratio. cos θ =Adjacent/Hypotenuse Let's substitute the corresponding values into this ratio and solve for a — the length of the longer leg of the triangle — and the apothem of the polygon.

cos θ =Adjacent/Hypotenuse

SubstituteValues

Substitute values

cos 60^(∘)=a/4

MultEqn

LHS * 4=RHS* 4

4cos 60^(∘)=a

RearrangeEqn

Rearrange equation

a=4cos 60^(∘)

We found that the apothem is 4cos 60^(∘)m. We already found that the perimeter is about 20.8m. To find the area, we will substitute these values in the formula A= 12ap.

A=1/2ap

SubstituteII

a= 4cos 60^(∘), p= 20.8

A=1/2( 4cos 60^(∘))(20.8)

UseCalc

Use a calculator

A=20.8

The area of the polygon to the nearest tenth is 20.8m^2.