Polygons
Take a minute and consider the following:
 What is a polygon?
 Are there any in this room?
 How are they used or are important?
Write your thoughts in your notebook or discuss them with a neighbor.
What are our thoughts on these questions?
Definitions:
 Polygon
o A shape made up of three or more straight lines that are
connected to make a closed space
 Regular Polygon
o A polygon where all the sides and angles are equal
 Irregular Polygon
o A polygon where not all the sides and angles are equal
 Diagonal
o A line connecting two corners of a polygon, but is not a side
of the polygon
Equations:
 Sum of all Angles of a Polygon
𝑆 = 180°(𝑛 − 2)
o S = sum
o n = number of sides
 Central Angle of a Polygon
360°
𝐶=
𝑛
o C = one central angle
o n = number of sides
 Number of Diagonals in a Polygon
𝑛(𝑛 − 3)
𝐷𝑖𝑎𝑔𝑜𝑛𝑎𝑙𝑠 =
2
o n = number of sides
Let’s take a moment and do our best to recall what we know about
polygons. Can you…




Name any?
Tell us the number of sides?
Tell us what the polygon’s angles add up to?
Tell us what one angle in a regular polygon would be?
Below is a chart of the names of the most common polygons we might
encounter.
Regular
Polygon
Triangle
Square
(quadrilateral)
Pentagon
Hexagon
Heptagon
Octagon
Nonagon
Decagon
Undecagon
Dodecagon
# of Sides
Sum of
Angles
Single Angle
# of
Diagonals
Let’s have a look at the handout I have that lists some of the properties
of quadrilaterals…
Examples:
 The sum of the interior angles of a polygon is 1260˚. Determine
the number of sides of the polygon.
o To do this I need to use the equation we copied down
earlier.
S = 180(n – 2)
o All I need to do is put the 1260˚ in for S and then work
backwards to figure out the number of sides, n.
1260 = 180(n – 2)
o The 180 is a problem since it is multiplying the brackets so
I’ll get rid of it by dividing by 180 on both sides of the equal
sign.
1260
180
=
180(𝑛−2)
180
7=n–2
o Now all I need to do is get rid of the 2 by adding 2 to both
sides of the equal sign.
2+7=n–2+2
n=7+2=9
o So this polygon has 9 sides (a nonagon).
 State the size of a central angle of a 9 sided regular polygon.
o All I need is to use the central angle formula and put in the
number of sides.
𝐶=
360°
𝑛
=
360
9
= 40°
o So a central angle in this polygon would be 40˚.
 Determine the number of diagonals in an 8-sided shape.
o All I need is to use the diagonals formula and put in the
number of sides.
𝐷𝑖𝑎𝑔𝑜𝑛𝑎𝑙𝑠 =
𝑛(𝑛−3)
2
=
8(8−3)
2
=
8(5)
2
=
40
2
= 20
So there are 20 unique diagonals in an 8-sided shape.
 State two properties that prove a quadrilateral is a parallelogram.
o To do this I need to look at my properties list in the handout.
o There are three facts on that sheet that define a
parallelogram.
 Both sets of opposite sides of the quadrilateral would
need to be parallel.
 Both sets of opposite sides of the quadrilateral would
need to be equal lengths.
 Both sets of opposite interior angles of the
quadrilateral would need to be equal.
Let’s try a few together:
 The sum of a polygon’s interior angles is 1620˚. How many sides
does it have? (11 sides)
 A regular polygon has 7 sides. How large is each of its interior
angles? (128.6˚)
 How large is one central angle in a 14-sided shape? (25.7˚)
 How many diagonals are in a 21-sided shape? (189)
 How many regular hexagons could fit around a single point? (Hint:
How many degrees make up a full circle?) (3 hexagons)
Practice
Complete the Polygons Practice handout.
Name:_______________ Date:_______________
Polygons Practice
1. Explain the difference between a regular polygon and an irregular
polygon.
2. Determine the number of sides for each regular polygon given the
following sums of the interior angles:
a. 180˚
b. 1080˚
c. 1440˚
3. Determine each of the following:
a. The central angle of a 7 sided regular polygon.
b. The number of sides a regular polygon would have if each of
the central angles is approximately 32.7˚.
4. A window frame is the shape of a regular octagon. What is the
interior angle between the frame’s pieces?
5. How many diagonals are in a 19-sided shape?
6. A gazebo is being built using a regular hexagon for the base. The
distance measured diagonally from one corner to another 10 ft.
10 ft
How long will each outside edge of the gazebo be?