Lesson Notes
The Area of a Regular Polygon
This section will focus on how to calculate
the Area of a Regular Polygon.
Regular Hexagon
We recall that a polygon is regular if all of its
sides are congruent and if all of its interior
angles are congruent.
Remember that congruent means identical.
In earlier sections we have calculated the area
of a triangle and the area of a square.
“a” is the length of the apothem
In this section, we will focus on the most
common regular polygons that use the length
of their apothem to calculate their area.
Regular Octagon
Note that for an equilateral triangle and a
square, it is possible to calculate their areas
without using an apothem.
Remember that a regular polygon with a
center O is made up of congruent isosceles
triangles. The altitude of each isosceles
triangle from the center point O is called the
apothem.
“a” is the length of the apothem
Regular Pentagon
In order to calculate the area of any of the
above regular polygons, we will use the
formula:
A=
( P )( a )
2
“A” is the area of the regular polygon
“P” is the perimeter of the regular
polygon
“a” is the length of the apothem
“a” is the apothem of the regular polygon
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1

Lesson Notes
Remember that the formula to calculate the
perimeter of a regular polygon is:
Now, we can state the formula for calculating
the area of a regular pentagon.
P = ( n )( b )
A=
( P )( a )
2
“P” is the perimeter
“n” is the number of sides
Finally, we can calculate the area (A).
“b” is the side length
A=
Can you see where we derived a formula
for the area of a regular polygon? Hint:
Remember that a regular polygon is
made up of congruent isosceles triangles.
2
( P )( a )
A = (100cm )(14cm )
A= 2
2
(100cm )(2 14cm )
A
=
1400cm
NoteAthat
= cm2
= (cm)(cm)
2
2
1400cm 2
A=
A = 7002cm 2
If you cannot think of how this formula,
A=
( P )( a )
( P )( a ) , came to be, ask your teacher to
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explain it to you in class.
Let’s try an example using this formula.
A = 700cm 2
Calculate the area of the regular polygon
given below.
So, the area of the regular pentagon is
700cm2.
Let’s try a different type of question where
we are given the area of a regular polygon
and we want to calculate the perimeter.
First, we state the information that we know
about the regular pentagon.
a = 14cm
b = 20cm
Area = 32.4cm 2
n=5
Next, we will calculate the perimeter (P) of
the regular pentagon.
First, we state the information that we know
about the regular hexagon.
P = ( n )( b )
Area = 32.4cm 2
a = 3cm
P = ( 5 )( 20cm )
P = 100cm
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2

Lesson Notes
Next, we state the area formula for a regular
hexagon.
A=
Example
( P )( a )
1.
2
Now, we can calculate the perimeter (P).
A=
Step 1: State the information you know
about the regular pentagon.
( P )( a )
P = 30cm
2
32.4cm 2 ( P )( 3cm )
=
1
2
2
( 2 ) 32.4cm = (1)( P )( 3cm )
(
A=
64.8cm 2 = ( 3cm )( P )
64.8cm 2 ( 3cm )( P )
=
3cm
3cm
( P )( a )
2
Step 3: Calculate the area (A).
A=
( 3cm )( P )
3cm
A=
cm
cm 2
=1
Remember that
= cm and that
cm
cm
( P )( a )
2
( 30cm )( 4cm )
2
120cm 2
A=
2
P = 21.6cm
A = 60cm 2
The perimeter of the regular hexagon is
21.6cm.
The area of the regular pentagon is
60cm2.
We are ready to do more challenging
examples.
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a = 4cm
Step 2: State the formula to calculate
the area of a regular pentagon.
)
21.6cm =
A regular pentagon has a perimeter
30cm and an apothem of 4cm. Calculate
the pentagon’s area.
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Lesson Notes
2.
3.
Calculate the area of the regular
hexagon given below.
A regular octagon has a side length
of 15cm and an apothem of 18cm.
Calculate the octagon’s area.
Step 1: State the information you know
about the regular octagon.
b = 15cm
a = 18cm n = 8
Step 2: Calculate the perimeter (P) of
the regular octagon.
Step 1: State the information you know
about the regular hexagon.
b = 6cm
a = 5cm
P = ( n )( b )
P = ( 8 )(15cm )
n=6
P = 120cm
Step 2: Calculate the perimeter (P) of
the regular hexagon.
Step 3: State the formula to calculate
the area of a regular octagon.
P = ( n )( b )
P = ( 6 )( 6cm )
A=
P = 36cm
Step 3: State the formula to calculate
the area of a regular hexagon.
A=
2
A=
Step 4: Calculate the area (A).
A=
( P )( a )
( P )( a )
2
(120cm )(18cm )
2
2160cm 2
A=
2
2
( 36cm )( 5cm )
A = 1080cm 2
2
The area of the regular octagon is
1080cm2.
180cm 2
A=
2
A = 90cm 2
The area of the regular hexagon is
90cm2.
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Step 4: Calculate the area (A).
( P )( a )
A=
A=
( P )( a )
4

Lesson Notes
4.
5.
Calculate the perimeter of the regular
pentagon given below.
Calculate the apothem of the regular
hexagon given below.
Area = 240cm 2
Area = 3240cm 2
Step 1: State the information you know
about the regular pentagon.
Area = 240cm
2
Step 1: State the information you know
about the regular hexagon.
a = 8cm
Area = 3240cm 2 b = 36cm n = 6
Step 2: State the formula to calculate
the area of a regular pentagon.
Step 2: Calculate the perimeter (P) of
the regular hexagon.
( P )( a )
A=
2
P = ( n )( b )
Step 3: Calculate the perimeter (P).
P = ( 6 )( 36cm )
( P )( a )
A=
P = 216cm
2
240cm 2 =
( P )(8cm )
Step 3: State the formula to calculate
the area of a regular hexagon.
2
240cm 2 = ( P )( 4cm )
A=
240cm 2 ( P )( 4cm )
=
4cm
4cm
60cm =
( P )( 4cm )
4cm
P = 60cm
The perimeter of the regular pentagon is
60cm.
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( P )( a )
2

Lesson Notes
Step 4: Calculate the apothem (a).
A=
3240cm 2 =
( P )( a )
2
( 216cm )( a )
2
3240cm 2 = (108cm )( a )
3240cm 2 (108cm )( a )
=
108cm
108cm
30cm =
(108cm )( a )
108cm
a = 30cm
The apothem of the regular hexagon is
30cm.
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student with an active Math Help Services account
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