Geometry
Areas of Regular Polygons
Goals



April 9, 2015
Find the area of equilateral
triangles.
Know what an apothem is and be
able to find its length.
Use the apothem to find the area of
a regular polygon.
Quick Review



April 9, 2015
30-60-90 Triangles
Right Triangle Trigonometry
Area of a triangle
30-60-90 Triangle
30
2a
a 3
60
a
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Trig Ratio Definition: Tangent
Opposite
A
Adjacent
Opposite
Tangent of A = Adjacent
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Area of any Triangle
A 
h
b
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1
2
bh
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Area of an Equilateral Triangle
A 
s
h
?
s
base
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s
1
2
bh
Finding h.
s
?s
1
2
April 9, 2015
h
s
s
We can solve for h by
using the Pythagorean
Theorem.
Finding h.
2
h 
2

h 
s
1
2
April 9, 2015
h
s
s
1
2
2
2
s
2
2
h
2
 s 
h
2

s  s
1
4
 s
2
h 
3
4
s
3
4
1
4
s
2
2
s
2

3
2
s
Solving for Area
1
2

1
2
s
s
3
2
s
April 9, 2015
A 
s

bh
s
3
4

3
2
s
2
s

Area of an Equilateral Triangle
s
s
s
April 9, 2015
A 
3
4
s
2
Example
Find the area.
Solution:
8
8
8
A 


3
4
3
4
s
2
 
8
2
3 16
64 
4
 1 6 3  2 7 .7
April 9, 2015
Your Turn
Find the area.
A 
10
10

10
April 9, 2015
3
4
3
4

10
2

25
1 0 0 
 2 5 3  4 3 .3
Example 2
The area of an
equilateral triangle
is 15. Find the
length of the sides.
A 
15 
3
4
3
4
s
2
s
2
 4  3 2
s
1 5   

3
 3  4
4
5.89
5.89
3 4 .6 4  s
5.89
s 
2
3 4 .6 4
s  5 .8 9
April 9, 2015
Area of a Regular Hexagon
Divide the hexagon
into six equilateral
triangles.
Each triangle has an
area of
s
April 9, 2015
A 
3
4
s
2
Area of a Regular Hexagon
Multiply this by 6:
A  6
s
April 9, 2015
A 
3 3
2
3
4
s
2
s
2
Example
Find the area of a
regular hexagon with
side length of 8.
A 

8
3 3
2
3 3
2
8
32
 64
 3 3  32
 96 3
 1 6 6 .3
April 9, 2015
2
Segments in a regular polygon.
Center
Radius
April 9, 2015
Apothem
Apothem


April 9, 2015
The perpendicular distance from the
center of a regular polygon to one of its
sides is called the apothem or short
radius. It is the same as the radius of a
circle inscribed in the polygon.
Apothem is pronounced with the
emphasis on the first syllable with the a
pronounced as in apple (A-puh-thum).
Apothem
Radius
April 9, 2015
Apothem
Another Way to Find the Area
The area of the
hexagon is equal to the
area of one triangle
multiplied by the
number of triangles, n.
Area = (Area of one )  (Number of s)
April 9, 2015
Area of one triangle
Radius
r
Apothem
a
s
April 9, 2015
A 
1
2
bh
A 
1
2
sa
This is the Area of
only one triangle.
Area of one triangle
Remember, there are
n triangles.
The total area then is
1
2
A 
r
a
s
April 9, 2015
sa  n
The perimeter of the
hexagon is s  n.
Perimeter
s
s
s
s
r
a
s
April 9, 2015
p=sn
s
A 
1
2
sa  n

1
2
a sn

1
2
ap
Area of a Regular Polygon
A 
1
2
a = apothem
p = perimeter
April 9, 2015
ap
This formula
works for all
regular polygons
regardless of the
number of sides.
Example
Find the area.
1. Draw a radius and an
apothem.
2. What kind of triangle
is formed?
12
a
r
60
6
x
30-60-90
3. What is the length of
the segment marked
x?
6
April 9, 2015
Example
Find the area.
4. So what is r?
12
5. And what is a?
12
12
r
6
April 9, 2015
a 3
6
6 3
6. The perimeter is?
72 (6  12)
Example
Find the area.
The apothem is
6 3
and the perimeter is 72.
The area is
12
12
6 3
A 
1
2
ap

1
2
6 3  7 2 
 216 3
 3 7 4 .1 2
April 9, 2015
Universal Formula
Click Here
to
Skip
April 9, 2015
Another Very Useful Formula

Given the length of a side, s, of a regular
polygon with n sides:
A


April 9, 2015
ns
2
4 tan 180 / n 
n = the number of sides
s = the length of a side
Previous Example Again
A
12

ns
2
4 tan  180 / n 
6  12
2
4 tan(180 / 6)

864
4 tan 30

864
2.3094
(graphing calculator)
April 9, 2015
 374.12
Notice!
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

April 9, 2015
In a regular hexagon, the radius is
always equal to the length of a side.
This is because we divide the
hexagon into equilateral triangles.
A hexagon is the only shape where
this is true.
The Fly in the Ointment…


April 9, 2015
If the polygon is anything other
than an equilateral triangle, a
square, or a regular hexagon,
finding the apothem and the radius
can be very challenging.
Use what you know about 30-60-90
triangles, 45-45-90 triangles, and
even trig to solve the problem.
A harder example
Find the area of the regular pentagon.
Where did 36 come from?
36
6
April 9, 2015
360
Each central angle
measures 1/5 of 360, or
72.
The apothem bisects the
central angle. Half of 72
is 36.
A harder example
Find the area of the regular pentagon.
What is the apothem?
6
36
6
What is the perimeter?
Don’t know.
Let’s find it.
April 9, 2015
A harder example
Find the area of the regular pentagon.
What trig function can be
used to find x?
36
(SOHCAHTOA)
6
Equation:
x
April 9, 2015
TANGENT
ta n 3 6 
x
6
A harder example
Solve the equation:
ta n 3 6 
36
6
6 ta n 3 6  x
6(.7 2 6 5)  x
x
Use a scientific calculator or
use the table on page 845.
April 9, 2015
x
6
x  4 .3 6
A harder example
x = 4.36
One side of the
pentagon measures?
36
8.72
6
The perimeter is
4.36
April 9, 2015
8.72 (2  4.36)
43.59 (5  8.72)
A harder example
The area is:
36
8.72
6
1
2
ap

1
2
 6   4 3 .5 9 
 1 3 0 .7 8
x
April 9, 2015
A 
Final Example
Find the area of a regular octagon
if the length of the sides is 10.
April 9, 2015
Step 1

Draw a regular octagon with side
length 10.
10
April 9, 2015
Step 2

Locate the center and draw a central
angle.
10
April 9, 2015
Step 3

Determine the measure of the
central angle.
360
8
10
45
April 9, 2015
 45
Step 4

Draw the apothem.
10
45
April 9, 2015
Step 5

The apothem bisects the angle and
the side. Write their measures.
10
22.5
45
5
April 9, 2015
Step 6

Use a trig function to find the
apothem.
ta n 2 2 .5 
10
a 
22.5
a
5
April 9, 2015
5
a
5
ta n 2 2 .5
a  1 2 .0 7
Step 7

Find the perimeter.
p = 10  8
p = 80
10
12.07
April 9, 2015
Step 8

Find the area.
p = 80
1
2
ap

1
2
1 2 .0 7   8 0 
 4 8 2 .8
A = 482.8
10
12.07
April 9, 2015
A 
Skip
This
Using the area formula:
A
10

ns
2
4 tan  180 / n 
8  10
2
4 tan(180 / 8)

800
4 tan 22.5

800
1.6569
 482.84
April 9, 2015
Summary


April 9, 2015
The area of any regular polygon can
be found be dividing the shape into
congruent triangles, finding the
area of one triangle, then
multiplying by the number of
triangles.
Or, multiply the length of the
apothem by the perimeter and
divide that by 2.
Practice Problems
April 9, 2015