7.1 Regular Polygons
Regular polygon: a polygon with all sides and all interior angles having the same measure.
Apothem: The apothem is a segment perpendicular to the sides of the polygon that connects the
center of the polygon with the middle of the sides that compose it.
Diagonal: it is a segment that connects two non-consecutive vertices of the polygon.
Interior angles of a regular polygon
For a regular polygon with n sides, the sum S of the inside angle measurements is :
𝑠 = (𝑛 − 2)×180!
𝑆𝑢𝑚 𝑜𝑓 𝑖𝑛𝑡𝑒𝑟𝑖𝑜𝑟 𝑎𝑛𝑔𝑙𝑒𝑠 = (𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑖𝑑𝑒𝑠 − 2)×180!
Examples :
1. What is the sum of interior angles for a pentagon?
𝑠 = 𝑛 − 2 ×180!
𝑠 = 5 − 2 ×180!
𝑠 = 3×180!
𝑠 = 540!
1. What is the sum of interior angles for a decagon?
𝑠 = 𝑛 − 2 ×180!
𝑠 = 10 − 2 ×180!
𝑠 = 8×180!
𝑠 = 1440!
3. How many sides does a regular polygon have if its sum of angles measures 1080o? Name that
polygon.
𝑠 = 𝑛 − 2 ×180!
1080 = 𝑛 − 2 ×180!
1080
= (𝑛 − 2)
180
6=𝑛−2
6+2=𝑛
8 = 𝑛 The polygon is an octagon.
The measurement a of an interior angle of a regular polygon is :
𝑎=
(𝑛 − 2)×180!
𝑛
For example :
What is the measure of an inner angle of the following heptagon:
𝑎=
(𝑛 − 2)×180!
𝑛
(7 − 2)×180!
7
5×180!
𝑎=
7
900
𝑎=
7
𝑎 = 128,57!
𝑎=
Example 2 :
An interior angle of a regular polygon measures 120o. How many sides does this polygon have?
120 =
(𝑛 − 2)×180!
𝑛
120𝑛 = 𝑛 − 2 180
120𝑛 = 180𝑛 − 360
120𝑛 − 180𝑛 = −360
−60𝑛 = −360
𝑛=6
The polygon has 6 sides and is a hexagon.
The perimeter of a regular polygon
The perimeter P of an n-sided regular polygon is :
P= 𝑛×𝑐
Where c designates the measurement on one side.
For example: c is the measurement on one side:
What is the measure of this hexagon?
P : 6×10 = 60 𝑐𝑚
The area of a regular polygon
The area A of a regular polygon is equal to
𝐴=
𝑐×𝑎×𝑛
2
Example :
The area of the regular pentagon or c=8 cm and a = 3 cm is :
𝐴=
5×8×3
= 60𝑐𝑚!
2