Area and Perimeter:
Areas of Regular Polygons
Review: Inscribed Polygons
& Circumscribed Circles
Inscribed means written inside
Circumscribed means written around (the outside)
inscribed polygon
circumscribed circle
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.
Def: A ________________is a polygon that is equiangular &
equilateral.
Inscribed Regular Polygons & Triangles
Inscribed Regular Pentagon
Total of Interior Angles = ___________
Each Interior Angle = ______________
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
Given: an inscribed regular n-gon (shown as an octagon)
A(n-gon) =
1
 n sa
2
1
 a(ns)
2
O
a
X
s
Y
Regular Polygon Area Theorem: The area of a regular
polygon is ___________________________________________
____________________________________________________
Regular Polygon Terminology
O
X
(Regular Octagon)
M Y
_______________________- the center of the circumscribed circle (O).
_______________________- the distance from the center to a vertex (OX).
_____________________________- an angle formed by 2 radii drawn
to consecutive vertices. ( XOY )
____________________________- the (perpendicular) distance from
the center of the polygon to a side. (OM)
Example: Square
r = 8 2 . Find a, p, A.
r
45
x
a
hyp  leg 2
p  ns
8 2 a 2
p  4(2x)
xa8
p  4[(2)(8)]
s
1
ap
2
1
 8(64)
2
A
Example: Equilateral Triangle
a = 4. Find r, p, A .
hyp  2short
r
30
p  ns
p  3(2x)
a
long  3 short
x
s
x  3(4)
A
1
ap
2
Example: Regular Hexagon
a = 5 3 . Find r, p, A.
long  3 short
a 3x
r
a
s
 6(2x)
 6(2)(5)
 60
60
x
p  ns
hyp  2 short
r  2(5)
r  10
A
1
ap
2
Regular Nonagon
r = 10; Find a, p, A.
a
r
a = 10(.9397)
a  9.397
X x
s
70
opp
sin X 
hyp
a
sin 70 
10
p  ns
 9(2x)
cos X 
adj
hyp
x
cos 70 =
10
x = 10(.3420)
x = 3.420
p = 9(2)(3.420)
p = 61.56
1
A  ap
2
1
 (9.397)(61.56)
2
= 289.24
Examples
r
r
r
a
a
A
1. 8 2
2.
8
5.
5
6.
49
3.
4.
r
6
a
a
p
6
4
7.
12
8.
9 3
A
More Examples
s
r
a
x
1. r = 4 2 , find A.
2. a = 6, find A.
r
a
x
3. a = 8, find p.
4. r = 12, find s.
r
a
x
5. s = 8, find r.