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This pdf contains the area of a regular polygon which is one-half the product of its apothem and its perimeter. Often the formula is written like this: Area=1/2(ap), where a denotes the length of an apothem, and p denotes the perimeter

1.
Areas of Regular Polygons

2.
Lesson Focus

The focus of this lesson is on applying the

formula for finding the area of a regular

The focus of this lesson is on applying the

formula for finding the area of a regular

3.
Basic Terms

Center of a Regular Polygon

the center of the circumscribed circle

Radius of a Regular Polygon

the distance from the center to a vertex

Central Angle of a Regular Polygon

an angle formed by two radii drawn to

consecutive vertices

Apothem of a Regular Polygon

the (perpendicular) distance from the center

of a regular polygon to a side

Center of a Regular Polygon

the center of the circumscribed circle

Radius of a Regular Polygon

the distance from the center to a vertex

Central Angle of a Regular Polygon

an angle formed by two radii drawn to

consecutive vertices

Apothem of a Regular Polygon

the (perpendicular) distance from the center

of a regular polygon to a side

4.
Basic Terms

5.
Theorem 11-11

The area of a regular polygon is equal to

half the product of the apothem and the

The area of a regular polygon is equal to

half the product of the apothem and the

6.
Area of a regular polygon

The area of a regular polygon is:

A = ½ Pa

Area

Perimeter

apothem

The area of a regular polygon is:

A = ½ Pa

Area

Perimeter

apothem

7.
The center of circle A is:

A

B The center of pentagon

BCDEF is:

A

F C A radius of circle A is:

A AF

A radius of pentagon

BCDEF is:

G AF

E D An apothem of pentagon

BCDEF is:

AG

A

B The center of pentagon

BCDEF is:

A

F C A radius of circle A is:

A AF

A radius of pentagon

BCDEF is:

G AF

E D An apothem of pentagon

BCDEF is:

AG

8.
Area of a Regular Polygon

• The area of a regular n-gon with side lengths (s)

is half the product of the apothem (a) and the

perimeter (P), so

The number of congruent

triangles formed will be

A = ½ aP, or A = ½ a • ns. the same as the number of

sides of the polygon.

NOTE: In a regular polygon, the length of each

side is the same. If this length is (s), and there

are (n) sides, then the perimeter P of the

polygon is n • s, or P = ns

• The area of a regular n-gon with side lengths (s)

is half the product of the apothem (a) and the

perimeter (P), so

The number of congruent

triangles formed will be

A = ½ aP, or A = ½ a • ns. the same as the number of

sides of the polygon.

NOTE: In a regular polygon, the length of each

side is the same. If this length is (s), and there

are (n) sides, then the perimeter P of the

polygon is n • s, or P = ns

9.
More . . .

• A central angle of a regular polygon is an

angle whose vertex is the center and

whose sides contain two consecutive

vertices of the polygon. You can divide

360° by the number of sides to find the

measure of each central angle of the

polygon.

• 360/n = central angle

• A central angle of a regular polygon is an

angle whose vertex is the center and

whose sides contain two consecutive

vertices of the polygon. You can divide

360° by the number of sides to find the

measure of each central angle of the

polygon.

• 360/n = central angle

10.
Areas of Regular Polygons

Center of a regular polygon: center of the circumscribed circle.

Radius: distance from the center to a vertex.

Apothem: Perpendicular distance from the center to a side.

Example 1: Find the measure of each numbered angle. 3

2

360/5 = 72 ½ (72) = 36 L2 = 36 1 •

L1 = 72 L3 = 54

Area of a regular polygon: A = ½ a p where a is the apothem and p is the perimeter.

Example 2: Find the area of a regular decagon with a 12.3 in apothem and 8 in sides.

Perimeter: 80 in A = ½ • 12.3 • 80 A = 492 in2

Example 3: Find the area.

10 mm

A=½ap p = 60 mm •

LL = √3 • 5 = 8.66 a

5 mm

A = ½ • 8.66 • 60 A = 259.8 mm2

Center of a regular polygon: center of the circumscribed circle.

Radius: distance from the center to a vertex.

Apothem: Perpendicular distance from the center to a side.

Example 1: Find the measure of each numbered angle. 3

2

360/5 = 72 ½ (72) = 36 L2 = 36 1 •

L1 = 72 L3 = 54

Area of a regular polygon: A = ½ a p where a is the apothem and p is the perimeter.

Example 2: Find the area of a regular decagon with a 12.3 in apothem and 8 in sides.

Perimeter: 80 in A = ½ • 12.3 • 80 A = 492 in2

Example 3: Find the area.

10 mm

A=½ap p = 60 mm •

LL = √3 • 5 = 8.66 a

5 mm

A = ½ • 8.66 • 60 A = 259.8 mm2

11.
• But what if we are not given any angles.

12.
Ex: A regular octagon has a radius

of 4 in. Find its area.

First, we have to find the

67.5

o

x

apothem length.

a x

4 sin 67.5 cos 67.5

4 4

a

3.7 4cos67.5 = x

4sin67.5 = a

135o 3.7 = a 1.53 = x

Now, the side length.

Side length=2(1.53)=3.06

A = ½ Pa = ½ (24.48)(3.7) = 45.288 in2

of 4 in. Find its area.

First, we have to find the

67.5

o

x

apothem length.

a x

4 sin 67.5 cos 67.5

4 4

a

3.7 4cos67.5 = x

4sin67.5 = a

135o 3.7 = a 1.53 = x

Now, the side length.

Side length=2(1.53)=3.06

A = ½ Pa = ½ (24.48)(3.7) = 45.288 in2

13.
Last Definition

Central of a polygon – an whose

vertex is the center & whose sides

contain 2 consecutive vertices of the

polygon.

Y is a central .

Measure of a

360

central is: n Y

Ex: Find mY.

72o

Central of a polygon – an whose

vertex is the center & whose sides

contain 2 consecutive vertices of the

polygon.

Y is a central .

Measure of a

360

central is: n Y

Ex: Find mY.

72o

14.
Check out!

ygonregulararea.html

ygonregulararea.html