GPS Algebra
Polygons Notes
Solutions Interior Angles of Polygons
Investigation Questions:
Name: __________________
How can the sum of the interior angles of a polygon be found?
How can the measure of one interior angle of a regular polygon be found?
Part 1
Number of
sides
Number of
diagonals
from 1
vertex
Number of
triangles
Interior
angle sum of
the polygon
Triangle
3
0
1
180o
Quadrilateral
4
1
2
360o
Pentagon
5
2
3
540°
Hexagon
6
3
4
720°
Heptagon
7
4
5
900°
Octagon
8
5
6
1080°
Decagon
10
7
8
1440°
Dodecagon
12
9
10
1800°
n-gon
n
n–3
n–2
180°(n – 2)
Polygon
Sketch
In the last row of the table you should have developed a formula for finding the sum of the interior
angles of a polygon. Use this formula to find the sum of the interior angles of a 20-gon.
180(20 – 2) = 3240°
Part 2
Regular Polygon
Nonregular Polygon
1. Compare the two polygons shown above. How would you define a regular polygon and a nonregular polygon? Both of the figures are hexagons, but the regular hexagon is equilateral and
equiangular. A nonregular polygon does not have all sides or angles congruent.
2. What is the sum of the interior angles of any hexagon? 180(6-2) = 720°
3. What is the measure of one angle of a regular hexagon? 720/6 = 120°
4. If you know the sum of the interior angles of a regular polygon, how can you find the measure of
one of the interior angles? The sum of the interior angles divided by the number of sides gives the
measure of one interior angle of a regular polygon.
180(𝑛−2)
𝑛
5. Use the information from Part 1 to complete the table below:
Regular
Polygon
Interior
angle sum
Measure of
one interior
angle
Triangle
180o
60°
Quadrilateral
360o
90°
Pentagon
540°
108°
Hexagon
720°
120°
Heptagon
900°
900/7 ≈128.6°
Octagon
1080°
135°
Decagon
1440°
144°
Dodecagon
1800°
n-gon
180(n – 2)
150°
180(𝑛 − 2)
𝑛
2
Conclusions:
Write a formula to find each of the following:
1. The sum of the interior angles of a polygon. 180(n – 2)
2. The measure of one interior angle of a regular polygon.
180(𝑛−2)
𝑛
In Class Problems:
6. What is another name for a regular triangle? Equilateral/Equiangular
7. What is another name for a regular quadrilateral? Square
8. Three angles of a quadrilateral measure 98 o, 75 o, 108 o. Find the measure of the fourth angle.
360 – (98+75+108) = 360 – 281 = 79°
7. Each interior angle of a regular polygon measures 168 o. How many sides does the polygon
have? Set the answer equal to the formula and solve.
180(𝑛−2)
168 =
; Cross multiply. 168n = 180n – 360; 30 sides
𝑛
-12n = -360
n = 30
8. What is the sum of the interior angles of the polygon above? 30(168) = 5040°
9. Is the sum of the interior angles of a convex polygon the same as a nonconvex polygon?
Explain or show an example to justify your answer. Yes, the sum of the interior angles is still
180°. There can only be two non-overlapping triangles formed by each polygon.
Nonconvex polygon
Convex polygon
3
Exterior Angles of Polygons
Questions:
What is the sum of the exterior angles of any polygon? 360°
How can the measure of one exterior angle of a regular polygon be found?
360°
𝑛
Regular
Polygon
Measure of one
interior angle
Number of
exterior
angles
Sum of the
exterior
angles
Measure of
one exterior
angle
Triangle
60°
3
360°
120°
Quadrilateral
90°
4
360°
90°
Pentagon
108°
5
360°
72°
Hexagon
120°
6
360°
60°
Heptagon
1287° ≈ 128.57°
7
360°
517° ≈ 51.43°
Octagon
135°
8
360°
45°
Decagon
144°
10
360°
36°
Dodecagon
150°
12
360°
30°
n-gon
180°(n – 2)
n
360°
360°
𝑛
4
4
3