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# of sides
# of triangles
Sum of measures of
interior angles
3 1 1(180)=180
4 2 2(180)=360
5 3 3(180)=540
6 4 4(180)=720
n n-2 (n-2) • 180
If a convex polygon has n sides, then the sum of the measure of
the interior angles is
(n – 2)(180°)
If a regular convex polygon has n sides, then the measure of one of the interior angles is
n
n 180)2(
Ex. 1 Use a regular 15-gon to answer the questions.
A)Find the sum of the measures of the interior angles.
B) Find the measure of ONE interior angle
2340°
156°
Ex: 2 Find the value of x in the polygon
130
126
143
100
117
x
126 + 130 + 117 + 143 + 100 + x = 720
616 + x = 720
x = 104
Ex: 3 The measure of each interior angle is 150°, how many sides does the regular polygon have?
n
n 180)2(One interior angle
150180)2(
n
n
nn 150180)2(
nn 150360180 36030 n
12nA regular dodecagon
Two more important terms
Exterior Angles
Interior Angles
The sum of the measures of the
exterior angles of a convex polygon, one
at each vertex, is 360°.
1
2
3
4
5
m m m m m 1 2 3 4 5 360
1
3
2
m m m 1 2 3 360
The sum of the measures of the
exterior angles of a convex polygon, one
at each vertex, is 360°.
1
3
2
4
m m m m 1 2 3 4 360
The sum of the measures of the
exterior angles of a convex polygon, one
at each vertex, is 360°.
The measure of each exterior angle of a regular polygon is
n
360
Ex. 4 Find the measure of ONE exterior angle of a regular 20-gon.
18°
sum of the exterior anglesnumber of sides
360
20
Ex. 5 Find the measure of ONE exterior angle of a regular heptagon.
51.4°
sum of the exterior anglesnumber of sides
3607
Ex. 6 The sum of the measures of five interior angles of a hexagon is 625. What is the measure of the sixth angle?
95°
Let’s practice!
11.1 Worksheet
s s
Area of an Equilateral Triangle
213
4A s
s
6060
30 30
Ex: 1 Find the area of an equilateral triangle with 4 ft sides.
213
4A s 24325.A
293.6 ftA
A Circle can be circumscribed around any regular polygon
VERTICES
A RADIUS joins the center of the regular polygon with any of the
vertices
A Central Angle is an angle whose vertex is the center and whose sides are two consecutive radii
n
360
A Regular HexagonEqual Angles
Equal Sides
How many equilateral triangles make up a regular Hexagon?
What is the area of each triangle?
What is the area of the hexagon?
s
213
4A s
6 • (the area of the triangle)
4
213
4A s The area of an equilateral triangle
The area of our equilateral triangle in this exampleA = 6.9282
The area of our hexagon in this exampleA = 6 * (6.9282)
How many identical equilateral triangles do we have? 6
41.569 units2 What is the area of this regular hexagon?
An APOTHEM is the distance between the center and a side. (It MUST be perpendicular to the side.)
How to find the
You need to know the apothem and perimeter
Area = (1/2)•a•P
or A = .5•a•P
Area of a Regular Polygon:
A = ½ aP
a
A = .5 (apothem) (# of sides)(length of each side)
A Regular Octagon
7 ft
45
7
3.5 ft
22.5°
x
360/8=45
7 ft
Perimeter is 56 feetApothem is 8.45 feet
What is the area?
Area = .5 • 8.45 • 56
Area = 236.6 ft2
11.2 WorksheetPractice B
ODDS
Worksheets’ EVENS
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