11.2 Areas of Regular Polygons
Geometry
Find the area of the triangle below
What if the sides were variables?
THEOREM 11.3
AREA OF AN EQUILATERAL TRIANGLE
The area of an equilateral triangle is one
fourth the square of the length of the side
times square root of 3.
Find the area of the equilateral
triangle
Parts of a Polygon
• The center of the polygon is the center of its
circumscribed circle
• The radius of the polygon is the radius of the
circumscribed circle
• The apothem of the polygon is the distance
from the center to any side of the polygon.
– The apothem is the height of a triangle in the
polygon
Finding the Area of a Regular Polygon
THEOREM 11.4
AREA OF A REGULAR POLYGON
The area of a regular n-gon with side length s
is half the product of the apothem a and the
perimeter P:
A = (1/2)aP
A = (1/2)a · ns
Find the area of the regular polygons
Examples
1. Find the area of an equilateral triangle with
10 cm sides.
2. Find the area of a regular nonagon with each
side being 8 and the apothem equal to 4.77
Central Angle of a Regular Polygon
A central angle of a regular polygon is an angle
whose vertex is the center.
You can divide 360 by the number of sides to
find the measure of each central angle of the
polygon.
Find the Central Angle
Find the Central Angle
When would we need to know the
central angle?
• Finding the side length
of the regular polygon
• Finding the apothem of
the regular polygon
REGULAR QUADRILATERAL
SQUARE
REGULAR HEXAGON
Examples
Examples
1. Find the area of a regular hexagon with each
side 5.
2. Find the area of a regular hexagon with a
radius of 8.
Let’s take a look at a couple of
regular polygons
POLYGON
QUADRILATERAL
PENTAGON
HEXAGON
HEPTAGON
# OF SIDES
CENTRAL ANGLE
BASE ANGLES
What about other regular
polygons?
Other Regular Polygons
We will have to use trig to find the missing sides of
other regular polygons.
1. Find the central angle
2. Draw in the apothem to make a right triangle
3. Divide the central angle by 2 to get the angle in the
right triangle
4. Set up a trig function to find the missing sides.
Find the area of the
Regular Pentagon
Find the area of the regular octagon
Find the Area of a Regular Dodecagon
The enclosure on the floor underneath the
Foucault Pendulum at Houston Museum of
Natural Sciences in Houston, Texas, is a regular
dodecagon with a side length of about 4.3 feet
and a radius of about 8.3 feet. What is the
floor area of the enclosure?
Example
The bottom of a glass is a regular 12-gon with
a side length of about 1.2 cm and a radius of
2.3 cm. What is the area of the bottom of the
glass?