Areas of Regular Polygons and
Circles
– Find areas of regular polygons.
– Find areas of circles.
AREAS OF REGULAR POLYGONS
First, some definitions:
Regular Polygon – a polygon in which all
segments and all angles are congruent.
Center of a Polygon – the center of its
circumscribed circle
Radius of a polygon – the radius of its
circumscribed circle, or the distance from the
center to a vertex.
Apothem of a polygon – distance from the
center to any side of the polygon.
AREAS OF REGULAR POLYGONS
Example:
Regular hexagon ABCDEF
Center and radius
Apothem
B
C
A
D
F
E
AREAS OF REGULAR POLYGONS
Example: regular hexagon
Notice that triangle GFA is
isosceles since all of the
radii are congruent.
The area of the hexagon
can be determined by
adding the areas of the
triangles.
B
C
G
A
F
D
E
AREAS OF REGULAR POLYGONS
Example: regular hexagon
Since the apothem is
perpendicular to the side
of the hexagon, it is an
altitude to ∆AGF
Area of ∆AGF = ½ ba
Area of the hexagon is 6(½ ba)
B
A
C
G
D
a
b
F
E
AREAS OF REGULAR POLYGONS
Example: regular hexagon
Notice that the perimeter P
of the hexagon is 6b units.
We can substitute P for 6b
in the area formula.
Area of the hexagon is 6(½ ba)
Area of the hexagon is ½ Pa
B
A
C
G
D
a
b
F
E
Key Concept
Area of a Regular Polygon
If a regular polygon has an area of A square
units, a perimeter of P units, and an apothem of a
units, then
A = ½Pa
This formula can be used to find the area of
any regular polygon.
Example 1
Area of a Regular Polygon
Find the area of a regular
pentagon with a perimeter
of 40 centimeters.
K
J
L
P
Step 1:
The internal angles of the
pentagon add up to 360°,
so …
N
Q
M
Example 1
Area of a Regular Polygon
Find the area of a regular
pentagon with a perimeter
of 40 centimeters.
K
J
L
P
Step 1:
The measure of each angle
360°
Is
or 72°
5
36°
N
Q
M
PQ is the apothem of pentagon JKLMN. It bisects
NPM and is a perpendicular bisector to NM. So
MPQ is ½(72°) or 36°.
Example 1
Area of a Regular Polygon
Find the area of a regular
pentagon with a perimeter
of 40 centimeters.
K
J
L
P
Step 2:
8
36°
Since the perimeter is 40
centimeters, each side is 8
centimeters and QM is 4
centimeters.
N
Q
4
M
Example 1
Area of a Regular Polygon
K
Write a trigonometric ratio
to find the length of PQ
J
L
P
QM
tan MPQ 
PQ
4

tan 36 
PQ
( PQ ) tan 36  4
4
PQ 
tan 36
PQ  5.5
8
36°
N
Q
4
M
Example 1
Area of a Regular Polygon
K
Area:
1
A  Pa
2
1
 (40)(5.5)
2
 110
J
L
P
8
5.5
N
Q
4
M
Key Concept
Area of a Circle
If a circle has an area of A square units and a
radius of r units, then
A = πr2
r
Example 2
Use Area of a Circle to
Solve Real World Problems
A caterer has a 48-inch
table that is 34 inches tall.
She wants a tablecloth that
will touch the floor. Find the
area of the tablecloth.
A  r 2
  (58) 2
 10,568.3
48
34
Example 3
Area of an Inscribed polygon
Find the area of the shaded
region. Assume the triangle
is equilateral.
4