Areas of Regular Polygons
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 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
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 polygon
The area of a regular polygon is:
A = ½ Pa
Area
Perimeter
apothem
B
F
A
G
E
D
The center of circle A is:
A
The center of pentagon
BCDEF is:
A
C A radius of circle A is:
AF
A radius of pentagon
BCDEF is:
AF
An apothem of pentagon
BCDEF is:
AG
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.
The number of congruent
triangles formed will be
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
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 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.
L2 = 36
½ (72) = 36
360/5 = 72
L1 = 72
3
2
1
•
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.
A = ½ • 12.3 • 80
Perimeter: 80 in
A = 492 in2
Example 3: Find the area.
10 mm
A=½ap
•
p = 60 mm
LL = √3 • 5 = 8.66
A = ½ • 8.66 • 60
a
A = 259.8 mm2
5 mm
• But what if we are not given any angles.
Ex: A regular octagon has a radius
of 4 in. Find its area.
67.5o
x
4
a
3.7
135o
First, we have to find the
apothem length.
a
sin 67 .5 
4
x
cos 67 .5 
4
4cos67.5 = x
4sin67.5 = a
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
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 mY.
360/5=
72o
Check out!
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ygonregulararea.html