Areas of Regular PolygonsAreas of Regular Polygons

Section 11.6Section 11.6

Lesson Focus

The focus of this lesson is on applying the formula for finding the area of a regular polygon.

Basic Terms

Center of a Regular Polygonthe center of the circumscribed circle

Radius of a Regular Polygonthe 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 Polygonthe (perpendicular) distance from the center of a regular polygon to a side

Basic Terms

Theorem 11-11

The area of a regular polygon is equal to half the product of the apothem and the perimeter.

Area of a regular polygonArea of a regular polygon

The area of a regular polygon is: The area of a regular polygon is:

A = ½ PaA = ½ PaArea Area

Perimeter Perimeter

apothemapothem

The center of circle A is:The center of circle A is:

AA

The center of pentagon The center of pentagon BCDEF is:BCDEF is:

AA

A radius of circle A is:A radius of circle A is:

AFAF

A radius of pentagon A radius of pentagon BCDEF is:BCDEF is:

AFAF

An apothem of pentagon An apothem of pentagon BCDEF is:BCDEF is:

AGAG

BB

CC

DDEE

FF

AA

GG

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

A = ½ aP, or A = ½ a • ns.

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 number of congruent triangles formed will be the same as the number of sides of the polygon.

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

Areas of Regular PolygonsCenter 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.

•123

360/5 = 72 ½ (72) = 36 L2 = 36

L3 = 54L1 = 72

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

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

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

Example 3: Find the area. 10 mm

•A = ½ a p p = 60 mm

5 mm

LL = √3 • 5 = 8.66 a

A = ½ • 8.66 • 60 A = 259.8 mm2

• But what if we are not given any angles.

Ex: A regular octagon has a radius Ex: A regular octagon has a radius of 4 in. Find its area.of 4 in. Find its area.

First, we have to find the First, we have to find the apothem length.apothem length.

4sin67.5 = a4sin67.5 = a

3.7 = a3.7 = a

Now, the side length.Now, the side length.

Side length=2(1.53)=3.06Side length=2(1.53)=3.06

44

aa

135135oo

67.567.5oo

45.67sin

a

3.73.7

xx

45.67cos

x

4cos67.5 = x4cos67.5 = x

1.53 = x1.53 = x

A = ½ PaA = ½ Pa = ½ (24.48)(3.7)= ½ (24.48)(3.7) = 45.288 in= 45.288 in22

Last DefinitionLast DefinitionCentral Central of a polygon of a polygon – an – an whose whose

vertex is the center & whose sides vertex is the center & whose sides contain 2 consecutive vertices of the contain 2 consecutive vertices of the polygon.polygon.

Y is a central Y is a central ..

Measure of a Measure of a

central central is: is:

Ex: Find mEx: Find mY.Y.

360/5=360/5=

7272oo

YYn

360

Check out!Check out!

http://www.mathopenref.com/polhttp://www.mathopenref.com/polygonregulararea.htmlygonregulararea.html