Area and Perimeter:Areas of Regular Polygons

Review: Inscribed Polygons & Circumscribed Circles

Inscribed means written inside

Circumscribed means written around (the outside)

Def: A polygon is inscribed in a circle & the circle is circumscribed about the polygon when each vertex of the polygon lies on the circle.

inscribed polygoncircumscribed circle

Def: A ________________is a polygon that is equiangular & equilateral.

Inscribed Regular Polygons & Triangles

Total of Interior Angles = ___________Each Interior Angle = ______________

Inscribed Regular Pentagon

5 congruent isosceles triangles

Total of Central Angles = _________Each central angle = _____________

Parts of a Regular Polygon A stands for Area

A(nonagon) is the area of a regular 9-sided figure.

n is the number of sides of a regular polygon p is perimeter, r is radius, s is side a is apothem

____________ – The line segment from the center of a regular polygon to the midpoint of a side or the length of this segment.

Sometimes known as the ______________, or the radius of a regular polygon’s inscribed circle.

Regular Polygon Area Theorem

Regular Polygon Area Theorem: The area of a regular polygon is _______________________________________________________________________________________________

A(n-gon) =

=n1

2sa

=1

2a(ns)

YX s

O

a

Given: an inscribed regular n-gon (shown as an octagon)

Regular Polygon Terminology

_______________________- the center of the circumscribed circle (O).

_______________________- the distance from the center to a vertex (OX).

____________________________- the (perpendicular) distance from the center of the polygon to a side. (OM)

_____________________________- an angle formed by 2 radii drawn to consecutive vertices. ( )∠XOY

YX M

O (Regular Octagon)

Example: Square

A=12aphyp=leg 2

8 2 =a 2

p=ns

p=4(2x)

p=4[(2)(8)]

=1

28(64)45

x =a=8r a

r = . Find a, p, A.8 2

s

x

Example: Equilateral Triangle

A=12aphyp=2short p=ns

x = 3(4)

p=3(2x)

long= 3 short30

ra

s

a = 4. Find r, p, A .

x

Example: Regular Hexagon

A=12aplong= 3 short

a= 3 x

p=ns

=6(2)(5)

hyp=2 short

r =2(5)

=6(2x)

=6060

r =10

s

a = . Find r, p, A.5 3

r a

x

Regular Nonagon

A=12ap

a=9.397

sin X =opphyp

sin 70 =a10

p=ns

cosX =adjhyp

=1

2(9.397)(61.56)

=9(2x)

cos 70 =x10

70

a=10(.9397 ) =289.24

r = 10; Find a, p, A.

r a

x

s

X

x=10(.3420 )

x=3.420

p=9(2)(3.420 )

p=61.56

Examples

r a A

1.

2. 8 5

3. 49

4.

8 2

6

r a p A

5. 6

6. 4

7. 12

8. 9 3

ra

ra

More Examples

ra

ra

1. r = , find A.2. a = 6, find A.

4 2 3. a = 8, find p.4. r = 12, find s.

5. s = 8, find r.

x x

ra

s

x