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Section 6.1: Angles of Polygons.
1) Find the sum of the 4 angles of a square:
2) What is the sum of the angles of a triangle?
3) Is the sum of the interior angles of a triangle always the same value? How
do you know?
Objectives: Understand angle relations in a polygon.
Students will be able to:
Find the sum of the interior angles of any polygon?
Find the measure of one interior angle of a regular polygon?
Find the sum of
Question: What is a polygon?
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Polygons can be classified as concave and convex.
Convex Polygon:
A convex polygon is defined as a
polygon with all its interior angles
less than 180°. This means that all
the vertices of the polygon will point
outwards, away from the interior of
the shape. Think of it as a 'bulging'
polygon. Note that a triangle (3-gon)
is always convex.
Convex or Concave?
Concave polygon:
A concave polygon is a polygon that is not
convex. A polygon is concave if at least
one of its internal angles is greater than
180°. (Concave when there are "dents" or
indentations in it. That is where the internal
angle is greater than 180°)) A star is a
concave polygon.
Investigation:
Part 1: Finding the sum of the interior angles of a polygon
In this investigation you are going to discover an easier way to find the sum of
the interior angles of a polygon, by dividing a polygon into triangles.
F
1. Consider the quadrilateral to the right.
Diagonal EG is drawn. A diagonal
is a segment connecting a vertex with a
nonadjacent vertex.
5
E
6
1
The quadrilateral is now divided into two
triangles, Triangle DEG and Triangle FEG.
Angles 1, 2, and 3 represent the interior angles of
Triangle DEG and Angles 4, 5, and 6 represent the
interior angles of Triangle FEG.
 mۧ1 + m  2 + m  3 = _________
_________
D 2
m 4 + m 5 + m 6 =
2. Mark the interior angles of the triangles with a small arc.
m  1 + m  2 + m  3 + m  4 + m  5 + m  6 = _________
3. What is the relationship between the sum of the angles in the quadrilateral
and the sum of the angles in the two triangles?
YOUR TURN:
In the provided chart, you will:
1. Name the polygon.
2. Sketch the polygon.
3. Select one vertex.
4. Draw all possible diagonals from that vertex.
5. Determine the number of triangles formed.
6. Calculate the sum of the interior angles.
When you have completed the investigation, answer the follow up questions on
the next page.
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Part 1 Investigation follow-up questions :
1. What is the relationship between the number of triangles you can draw in
each polygon to the number of sides of the polygon?
1. How can we use the number of triangles in a polygon to find the sum of
the angles of that polygon? (Hint: recall the number of degrees in a
triangle)
2. Describe, in words, how we can use the information discovered in steps 1
and 2 to find the sum of interior angles of any polygon.
3. An n-gon is used to describe a polygon with n sides. Write a formula,
using n to represent the number of sides, that will allow you to find the
sum of the interior angles of an n-gon:
Sum of Interior Angles of an N-gon =
4. Find the sum of the interior angles in a 20-gon.
5. Can you find the value of one interior angle in any polygon?
Part 2: Finding the measure of a single interior angle
Regular hexagon
Nonregular hexagon
1. Compare the two polygons shown above. How would you define a regular
polygon and a nonregular polygon?
2. What is the sum of the interior angles of a hexagon?
3. What is the measure of one angle of a regular hexagon?
4. If you know the sum of the angles of a regular polygon, how can you find the
measure of one of the congruent angles?
5. Use the information from Part 1 to complete the table below:
Regular
Polygon
Interior
angle sum
Triangle
180o
Quadrilateral
360o
Pentagon
Hexagon
Heptagon
Octagon
Nonogon
Decagon
Dodecagon
n-gon
Measure
of one
angle
Part 3: Finding the sum of the exterior angles of a polygon.
Use the regular polygons to find the sum of the exterior angles.
Follow up questions:
1. What is the sum of the exterior angles of a triangle?
2. What is the sum of the exterior angles of a square?
3. What is the sum of the exterior angles of any polygon?
4. What is the sum of the exterior angles of a 100-gon?
5. Find the measure of a single exterior angle of a regular 20-gon?
6. Find the measure of a single exterior angle of a regular n-gon.
SUMMARY QUESTIONS:
Write the formula for
a. the sum of the interior angles of any polygon.
b. the measure of a single interior angle of a regular polygon.
c. The sum of the exterior angles in any polygon.
d. The measure of a single exterior angle of a regular polygon.