2.7
What is Its Measure?
Pg. 23
Interior Angles of a Polygon
2.7 – What is Its Measure?_____________
Interior Angles of a Polygon
In an earlier chapter, you discovered that
the sum of the interior angles of a triangle
is always 180°. But what about the sum
of the interior angles of other polygons,
such as hexagons or decagons? Does it
matter if the polygon is convex or not?
Consider these questions today as you
investigate the angles of a polygon.
2.36 – INTERIOR ANGLES
With your team, find the sum of the
measures of its interior angles as many
ways as you can. You may want to use
the fact that the sum of the angles of a
triangle is 180°. Be prepared to share
your method with the class.
54° 54°
54°
54°
54°
72° 72°
72° 72°
72°
54°
54°
54°
54°
54°
540°
d
b
a
f
180°
180°
g
180°
c e
h
i
540°
2.37 – SUM OF THE INTERIOR ANGLES OF
A POLYGON
In the previous problem, you found the sum
of the measures of the interior angles of a
pentagon. But what about other polygons?
a. Use your method from the previous
problem to find the sum of the interior
angles of other polygons. Draw a sketch
to help support your answers. Find the
sum of the measures of several shapes.
(It might be helpful to split up the tasks
within your group).
b. Complete the chart below based on your
findings.
# of sides
of polygon
Sum of
interior
angles
3
180°
4
5
6
360° 540° 720°
b. Complete the chart below based on your
findings.
# of sides
of polygon
Sum of
interior
angles
7
8
9
10
900° 1080° 1260° 1440°
b. Complete the chart below based on your
findings.
# of sides of
polygon
Sum of
interior
angles
15
2340°
25
50
4140° 8640°
c. What if it was has "n" number of sides?
What is the sum of the interior angles
then? Come up with a formula to find the
sum of the interior angles of any polygon.
180(n – 2)
where n = # of sides
2.38 – FINDING ANGLES
Use the angle relationships in each of the
diagrams below to solve for the given
variables. Show all work.
7
1
2
6
180(n – 2)
180(7 – 2)
900°
3
5
4
3m + 501 = 900
3m = 399
m = 133°
180(n – 2)
180(6 – 2)
720°
6x = 720
x = 120°
180(n – 2)
180(8 – 2)
1080°
32x + 824 = 1080
32x = 256
x = 8°
2.39 – USING INTERIOR AND EXTERIOR
ANGLES
Use your understanding of polygons to
answer the questions below, if possible. If
there is no solution, explain why not.
a. How many sides does a polygon have if
the sum of the measures of the interior
angles is 1980°? 2,520°?
a. How many sides does a polygon have if
the sum of the measures of the interior
angles is 1980°? 2,520°?
180(n – 2) = 1980 180(n – 2) = 2520
n – 2 = 14
n – 2 = 11
n = 16
n = 13
b. A quadrilateral has four sides. What is
the measure of each of its interior angles?
Depends on if it
is regular or not
c. What if you know that the measure of an
interior angle of a regular polygon is 162°?
How many sides must the polygon have?
each interior angle = 180(n – 2)
n
180(n – 2) = 162°
1
n
180(n – 2) = 162n
180n – 360 = 162n
–360 = –18n
20 = n
c. Each interior angle of a regular
pentagon has measure 2x + 4°. What is x?
Sum of pentagon:
180(5 – 2) = 540°
Each angle is 2x + 4, so all 5 add to….
5(2x + 4) = 540
10x + 20 = 540
10x = 520
x = 52°
2.40 – CENTRAL VS INTERIOR ANGLES
Below is a list of regular polygons. Find
the measure of each central angle and
each interior angle. Show all work.
180(n – 2)
180(8 – 2)
1080°
135°
45°
360/8 = 45°
1080/8 = 135°
108°
180(n – 2)
180(5 – 2)
540°
72°
360/5 = 72°
540/5 = 108°
180(n – 2)
180(40 – 2)
6840°
360/40 = 9°
6840/5 = 171°
2.41 – CONCLUSIONS
How do you find the sum of the interior
angles? How is this measure different than
the central angle?
Sum of the Interior
Angles
Each interior angle in
a regular polygon
180(n – 2)
180(n – 2)
n