G.GMD.1 Wkst 5

G.GMD.1 STUDENT NOTES WS #5
1
REGULAR POLYGONS
Regular polygons are of interest to us because we begin looking at the volume of a hexagonal prism or a
Tetrahedral and to do these types of calculations we need to be able to solve first for their areas. The term
regular refers to a very specific type of shape. It refers to the polygon that has all equal sides, all equal angles
and the maximum lines of symmetry for that particular number of sides.
Regular polygons can be inscribed in a circle and because of this we apply a few new terms to these polygons.
First of all, each regular polygon has a center and a radius because they are inscribed in the circle. The other
new term is the apothem. The apothem is the perpendicular distance from the center to a side of the regular
polygon. In the equilateral, square and hexagon the triangles that are formed by the radius, the apothem and
a side are all special right triangles. In the cases of the equilateral and hexagon it is a 30°- 60° right triangle
and in the case of the square a 45°- 45° right triangle is formed. These triangles are very important in helping
us determine the area of these regular polygons.
s
r
s
60 a
30
0.5s
45
r
a
30
r
a
45
0.5s
r
60
r
a
60
0.5s
45
30
a
r
a
30
0.5s
45
0.5s
s
60
0.5s
G.GMD.1 STUDENT NOTES WS #5
2
The way to visualize area of these regular polygons is to calculate the area of congruent triangles inside them.
We find the area of one triangle and then multiply the area by 3 (3 ≅ triangles) in the case of an equilateral or
by 6 (6 ≅ triangles) in the case of a hexagon. Let’s do a few examples so you can see the steps.
Given an equilateral with a radius of 6 cm, determine the area.
If the radius is 6 cm, then the short
leg (apothem) is 3 cm and the long
leg (½ side) is 3 3 , thus making
the side of the equilateral 6 3 .
6
60 3
30
3 3
So to determine the area it
would be:
Area = 3( Area of triangle)
6
1
Area = 3   (base)(height )
2
1
Area = 3   (6 3 )(3)
2
30
3 3
60 3
3 3
Area = 27 3 cm 2
The skills learned earlier when working with special right triangles come in very handy here!!
Given a regular hexagon with a side of 8 cm, determine the area.
If the side is 8 cm, then the short
So to determine the area it would
leg (½ side) is 4 cm and the long leg be:
(apothem) is 4 3 .
Area = 6 ( Area of triangle)
8 30
4 3
60
4
1
Area = 6   (base)(height )
2
1
Area = 6   (8)(4 3)
2
Area = 96 3 cm
r
30
4 3
8
60
4
2
These types of questions vary concerning what the given information is but they all resort back to the special
right triangle inside the equilateral triangle, the square or the regular hexagon. Two formulas that provide
shortcuts for area are derived when the side length is given.
Given the side, s, then the long leg is half of that. The
apothem is that value divided by 3 which
s 3
rationalizes to
. Now we are ready to calculate
6
the area:
1 s 3
Area = 3   ( s) 

 2   6 
Area =
s2 3
cm 2
4
s
60 s 3
r
6
30
s
s
2
2
G.GMD.1 STUDENT NOTES WS #5
3
Given the side, s, then the short leg is half of
that. The apothem is that valued multiplied
s 3
. Now we are ready to
by 3 which is
2
calculate the area:
30
s
s 3
r
1 s 3
Area = 6   ( s) 

 2   2 
2
60
3s 2 3
Area =
cm 2
2
s
2
Instead of leaving these two
Equilateral Formula
formulas as they are we could have
rearranged the information to
1 s 3
summarize them a little differently. Area = 3  2  ( s )  6 
  

When we do this we get another
s 3
way to think about the area of a
1
Area =   3( s ) 

regular polygon.
2
 6 
1
Area =   ( perimeter )(apothem)
2
AREAREGULAR POLYGON =
Hexagon Formula
1 s 3
Area = 6   ( s ) 

 2   6 
s 3
1
Area =   6( s ) 

2
 6 
1
Area =   ( perimeter ) ( apothem )
2
1
(perimeter)(apothem)
2
For all of the other regular polygons, trigonometry is needed because the angles within their triangles are not
the special ones. Let me solve one of these to demonstrate the steps.
Given a regular pentagon with side length of 8 cm, determine the area.
When you divide the shape into
5 congruent triangles we get
360/5 = 72° angle. When drop
the altitude it create the
apothem a 36° right triangle is
formed. So using trigonometry
we can determine the apothem
value.
4
tan 36° =
a
4
a=
tan 36°
Now we can determine the area of
the regular pentagon.
A = (5∆ )( Area of ∆)
1  4 
A = 5   (8) 

 2   tan 36° 
A ≈ 110.11 cm 2
72°
72°
72°
72°
36°
a
4 cm
4 cm
G.GMD.1 STUDENT NOTES WS #5
4
Find the AREA of each regular polygon.
Hexagon with radius 8 cm
_____________(E)
Square with an apothem of 6 cm
Equilateral triangle with radius 4 3 cm
_____________(E)
_____________(E)
4 3 cm
8 cm
2 3 cm
6 cm
30°
4 3 cm
30°
6 cm
6 cm
4 cm
A = ½ ap
A = ½ ( 4 3 )(8)(6)
A = 96 3 cm2
A = ½ ap
A = ½ (6)(4)(12)
A = 144 cm2
A = ½ ap
A = ½ ( 2 3 )(12)(3)
A = 36 3 cm2
G.GMD.1 WORKSHEET #5
NAME: ____________________________ Period _______
1. What is the central angle of a regular
EQUILATERAL
a) Octagon?
_______
b) Hexagon? _______
c) Decagon?
_______
d) Triangle?
_______
s
2. Find the apothem of each regular polygon.
r
60° a
30°
a) Hexagon with side of 15 cm
0.5s
60°
r
a
_____________(E)
b) Square with a diagonal of 12 2 cm
30°
0.5s
SQUARE
_____________
c) Equilateral triangle with side 24 cm
r 45°
a
0.5s
_____________(E)
45°
r
a
3. Find the radius of each regular polygon.
45°
a) Square with side 7 cm
0.5s
HEXAGON
_____________(E)
b) Equilateral triangle with side of 4 3 cm
r 30°
a
60°
0.5s
_____________
c) Hexagon with side of 8 cm
30°
r
60°
0.5s
_____________
a
1
G.GMD.1 WORKSHEET #5
4. Find the AREA of each regular polygon.
2
EQUILATERAL
a) Square with radius of 16 2 cm
s
r
60° a
30°
_____________
b) Equilateral triangle with an apothem of 2 cm
0.5s
60°
r
a
30°
0.5s
_____________(E)
c) Hexagon with a perimeter of 18 cm
SQUARE
r 45°
_____________(E)
d) Equilateral triangle with a radius of 16 3 cm
a
0.5s
45°
r
a
_____________(E)
e) Regular Octagon with side of 12 cm.
45°
0.5s
HEXAGON
r 30°
a
60°
0.5s
30°
_____________(2 dec.)
r
60°
0.5s
a
G.GMD.I WORKSHEET
#5
NAME:
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1. What is the central angle of a regular
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c) Hexagon with side of 8 cm
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G.GMD,7 WORKSHEET #5
4. Find
th<pof
each regular polygon.
a) Square with radius
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cm
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b) Equilateraltriangle with an apothem of 2 cm
_(E)
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e) Regular Octagon with side of 12 cm.
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