Exercise 13 Page 657

The area of a regular polygon is half the product of the apothem and the perimeter. Note that we are given the apothem and the radius.

We will first need to find the side length and the perimeter of the polygon. Then we will use the formula A= 12ap to find the area.

Finding the Perimeter

To find the perimeter, we first need the side length of the polygon. Notice that we have a right triangle formed by the radius and the apothem.

The apothem bisects any side length of the polygon. By solving the right triangle, we will have half of the side of the polygon. Let's use the Pythagorean theorem to find the value of l.

c^2=a^2+b^2

SubstituteValues

Substitute values

4^2= 3.3^2+ l^2

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Solve for a

CalcPow

Calculate power

16=10.89+l^2

SubEqn

LHS-10.89=RHS-10.89

5.11=l^2

SqrtEqn

sqrt(LHS)=sqrt(RHS)

±sqrt(5.11)=l

RearrangeEqn

Rearrange equation

l=±sqrt(5.11)

Let's take the positive root because l is a distance. Therefore, the value of l is sqrt(5.11). Moreover, the length of one side of the given regular polygon is 2* sqrt(5.11)= 2sqrt(5.11).

In a regular pentagon all five sides have the same length. Therefore, we can obtain its perimeter by multiplying the length of a side by 5. Perimeter: 2sqrt(5.11)* 5= 10sqrt(5.11)

Finding the Area

Finally, we have that the apothem is 3.3 and that the perimeter is 10sqrt(5.11). To find the area, we will substitute these two values in the formula A= 12ap and simplify.

A=1/2ap

SubstituteII

a= 3.3, p= 10sqrt(5.11)

A=1/2( 3.3)( 10sqrt(5.11))

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Evaluate right-hand side

Multiply

Multiply

A=1/2(33sqrt(5.11))

MoveRightFacToNumOne

1/b* a = a/b

A=33sqrt(5.11)/2

UseCalc

Use a calculator

A=37.29876...

RoundDec

Round to 2 decimal place(s)

A≈ 37.30

The area of the given regular polygon is about 37.30 square units.