5.8

What Is The Area?

Pg. 27Finding the Area of Regular Polygons

5.8 – What Is The Area?Finding the Area of Regular Polygons

In this chapter you have developed a method to find the measures of the sides of a right triangle. How can this be useful? Today you will use what you know about the angles of a regular polygon and right triangles to explore how to find the area of any regular polygon with n sides.

Center of a polygon:

Point equidistant to the vertices of the of the polygon

P

Radius of a polygon:

Length from the center to the vertex of a polygon

PM

PN

Apothem of the polygon:

Length from the center to the side of a polygon

PQ

Central angle of a regular polygon:

Angle formed by two radii in a polygon

MPN360

n

Find the given angle measure for the regular hexagon shown.

Each central angle =

360

n

360

6 60°

60°

60°

60°

m EGF

m EGD

Find the given angle measure for the regular hexagon shown.

30°

60°30°

m EGH

m DGH

30°

30°

m GHD 90°

5.40 – MULTIPLE STRATEGIESWith your team, find the area of each shape below. Make sure that your results from using different strategies are the same. Make sure everyone on your team agrees.

3604

= 90°

45°

4

A = ½bh

A = ½(8)(4)

4

A = 16x 4

A = 64un2

3605

= 72°

36°

tan 36° = 3a

a = 4.13

4.13

3

A = ½bh

A = ½(6)(4.13)

A = 12.39x 5

A = 61.95un2

5.41 – WRITING THE DIRECTIONSThe height of the triangle in a regular polygon is called an apothem (a-poth-um). Given the picture below, come up with a formula that will give you the area of the regular hexagon.

1

2A san

s = side length (base of triangle)a = apothem (height of triangle)

n = # of sides (# of triangles)

5.42 – GIVEN THE VALUESA regular pentagon has a side length of 8in and an apothem length of 5.5in. Find the area.  1

2A san

1(8)(5.5)(5)

2A

(4)(27.5)A

2110A in

5.43 – EXTRA PRACTICEFind the area of the two regular polygons below. Look for special triangles or SOH-CAH-TOA to help find the missing lengths.

360

660°

30°

5m5m60°

5 3

1

2A san

1(10)(5 3)(6)

2A

(5)(30 3)A

A 150 3m2

30°

5m5m

5 3

72°

36°

360

5

36°

O

Atan 36

x =15.98 15.9815.98

22

x

1

2A san

1(31.96)(22)(5)

2A

21757.8A cm

36°

15.9815.98O

A

3608

= 45°

22.5°

tan 22.5° = x9

x = 3.73

3.733.73

A = ½san

A = ½(7.46)(9)(8)

A = 268.56un2

5.44 – FERTILIZERBeth needs to fertilize her flowerbed, which is in the shape of a regular nonagon. A bag of fertilizer states that it can fertilize up to 150 square feet, but Beth is not sure how many bags of fertilizer to buy. Beth does know that each side of the nonagon is 16 feet long. Find the area of the flowerbed and tell Beth how many bags of fertilizer to buy.

16ft 88

3609

= 40°

20°

O

A

tan 20° = 8a

a = 21.98

A = ½san

A = ½(16)(21.98)(9)

A = 1582.56un2

1582.56/15011 bags

5.58 – CONCLUSIONSExamine the steps to find the area of the polygon below. How does this compare to your directions? 1. Find central angle (360/n)2. Divide by 2 for height splitting angle in half3. Find the base and/or height of triangle4. Find the area of the triangle5. Multiply by the number of triangles