11-2: Areas of Regular Polygons

Essential Question:

How are measurements of two-dimensional figures useful for modeling situations in the real world?

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Essential Skill(s):

I can compute(calculate) perimeter and area.

I can apply geometric methods to solve design problems utilizing real-world content.

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Formulas: a: apothem, P: Perimeter,

Regular Polygon: A = (aP)

n-gon the (n)number of sides

s = length of a side

a

1

2

s

Perimeter: P = ns

A regular polygon can be divided into little congruent triangles

P

a

Additional Rules;

Central angle = 360 ÷ n

Apothem is an angle bisector: (central angle) ÷ 2

Apothem is a perpendicular bisector: s ÷ 2

Radii of a Polygon: the distance from the center point to each vertex.

Can use: a² + b² = c² , special right triangles or trig ratios to find the apothem, radii, or half a side

Ex 1: Find the measure of each numbered angle.

1

2

3

1) Id the shape hexagon n = 6

2) Find central angle 1: ∠1 = 360 ÷ n

3) Find angle 2: ∠2 = ∠1 ÷ 2

4) Find complementary angle 3:

∠3 = 90 −∠2

∠1 = 360 ÷ 6 = 60

∠2 = 60 ÷ 2 = 30

∠3 = 90 − 30 = 60

Process:

• Identify the shape and number of sides
• Label a, s, n
• Evaluate Perimeter P=ns
• Evaluate Area A = ½(aP)

Ex #2: Find the perimeter and area. {given apothem and side length}

10 m

5 m

1) id shape heptagon n=7 sides

2) label apothem(a), side length(s)

3) evaluate P=ns P = (7)(10)

multiply P = 70m

s =

a =

4) Evaluate A=½(aP) A = (5)(70)

1

2

Multiply A = (350)

1

2

Simplify A = 175m²

Ex #3: Find the area of regular polygon given a side.

{Round to the nearest tenth}

s =

4ft

1) id shape hexagon n=6 sides

2) label side length(s) apothem(a)

3) Perimeter P = (6)(4)

4) Find area

Need angle measure {see example 1}

30

Find apothem *use special right triangles

a = 2 3

a =?

= 24ft

6) Find area A = (2 3 )(24)

1

2

30°

A = 24 3 ft²

41.6 ft²

s =

4ft

Find apothem *use special right triangles

{since the central angle is 60, equilateral triangle}

a² + b² = c²

a =?

6) Find area A = (2 3 )(24)

1

2

30°

A = 24 3 ft²

41.6 ft²

a² + 4 = 16

a² = 12

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a² = 12

a = 2 3

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a² + (2)² = (4)²

30°

2

{half the side length}

4ft

a =?

Ex #4: Find the perimeter and area. {given radius}

6 m

5 m

1) id shape pentagon n=5 sides

2) label apothem(a), side length(s)

3) evaluate P=ns P = (5)(6)

multiply P = 30m

s =

a =?

4) find apothem a²+b²=c²

a² + 3²= 5²

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a² + 9 = 25

a² = 16

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5) Evaluate A=½(aP) A = (4)(30)

1

2

a² = 16

a = 4

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A = (120)

1

2

A = 60m²

r =