Download Lesson 31-1: Sum of the Measures of the Interior Angles of Polygons

Lesson 31-1: Sum of the Measures
of the Interior Angles of Polygons
We will develop a formula for the sum of
the measures of the interior angles of a
polygon
and
I will determine the sum of the measures
of the interior angles of a polygon.
March 31, 2017
The sum of the angle measures of a triangle is
180º. We can use that to figure out the sum
of the measures of a convex polygon.
Sum of the s of a Quad. =
Sum of the s of two ∆s =
2(180º) = 360º
Pentagon - 3 s
3(180º) = 540º
Angle Sum Worksheet
Polygon
Triangle
Number of
Sides
Number of
Triangles
Sum of the
measures of the
interior angles
3
1
(1)180º = 180º
Quadrilateral 4
2
(2)180º = 360º
Pentagon
5
3
(3)180º = 540º
Hexagon
6
4
(4)180º = 720º
Heptagon
7
5
(5)180º = 900º
Octagon
8
6
(6)180º = 1080º
Nonagon
9
7
(7)180º = 1260º
N-gon
n
n-2
(n – 2)180
Polygon Interior Angles Theorem
The sum of the interior angles of an
n-gon is
(n - 2)(180)
Examples
Find the measure of angle x.
116º
72º
155º
x
148º
What do we need to know first?
Sum of the interior angles:
(n-2)(180) = (5 - 2)(180) = 3(180) = 540
155 + 116 + 72 + 148 + x = 540
491 + x = 540
x = 49
Example: The measures of the interior angles of a
polygon are related as shown. Find the measure of
each angle.
x + 20
x + 40
x
x + 80
x + 60
x  x  20  x  40  x  60  x  80  540
5x  200  540
5x  340
x  68
68º, 88º, 108º, 128º, 148º
Observe that as the number of sides increases by
one, the sum of the angle measures also increases by
a constant amount. What type of function models this
behavior?
linear
Sum of Int. 
For the first six polygons in the previous chart, plot
the ordered pair (number of sides, sum of angle
measures) on the axes below. Carefully choose and
label your scale
on each axis.
1260
1170
1080
990
900
810
720
630
540
450
360
270
180
90
The data points you graphed above should appear
collinear. Write an equation for the line determined
by these points.
(3, 180); (4, 360)
360  180
m
4 3
m  180
y  y1  m(x  x1 )
y  180  180(x  3)
y  180  180x  540
y  180x  360
y  180(x  2)
How does this compare to
what we found previously?
10. State the numerical value of the slope of the line
in Item 11 and describe what the slope value tells
about the relationship between the number of sides
and the sum of the measures of the interior angles of
a polygon. Use units in your description.
The numerical value of the slope of the line is 180.
This slope tells us that each time a side is added to a
polygon, the sum of the measures of the interior
angles of the polygon increases by 180°.
Whole numbers ≥ 3
The function forms a straight line.
Domain is restricted because
1. a polygon cannot be formed with fewer than 3
sides
2. it cannot have a fractional number of sides.