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Big Ideas Math Geometry, 2014

BI

Big Ideas Math Geometry, 2014 View details

Chapter Review

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Exercise 12 Page 657

Find the perimeter and the apothem. Then substitute their values in the formula A= 12ap to find the area of the regular polygon.

≈ 167.11square units

Practice makes perfect

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

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

Finding the Perimeter

Since we have a regular nonagon whose side length is 5.2, we can find the perimeter by multiplying 9 by 5.2.

9* 5.2= 46.8 The perimeter of the given polygon is 46.8.

Finding the Apothem

Let's now find the apothem. To do so, we will start by drawing the apothem of the nonagon.

The apothem is perpendicular to any side of the polygon and bisects it. As a result, we have a right triangle with a hypotenuse of 7.6. The length of its shorter leg is 5.2 ÷ 2=2.6. The height a is unknown.

To find the value of a, let's use the Pythagorean theorem.

c^2=a^2+b^2

c= 7.6, b= 2.6

7.6^2=a^2+ 2.6^2

▼

Solve for a

Calculate power

57.76=a^2+6.76

LHS-6.76=RHS-6.76

51=a^2

sqrt(LHS)=sqrt(RHS)

±sqrt(51)=a

Rearrange equation

a=±sqrt(51)

Because a is a measure of length, it needs to be the positive root. Therefore, the length of the apothem is sqrt(51).

Finding the Area

To find the area of the given regular polygon, we will substitute a=sqrt(51) and p= 46.8 in the formula A= 12ap.

A=1/2ap

a= sqrt(51), p= 46.8

A=1/2(sqrt(51))( 46.8)

▼

Evaluate right-hand side

Multiply

A=1/2(46.8sqrt(51))

MoveRightFacToNumOne

1/b* a = a/b

A=46.8sqrt(51)/2

Use a calculator

A=167.10942...

Round to 2 decimal place(s)

A≈ 167.11

The area of the given regular polygon is about 167.11 square units.