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Name of Lecturer: Mr. J.Agius
Course: HVAC1
Lesson 50
Chapter 9: Angles and Shapes
Interior Angles of Polygons
An Interior Angle is an angle inside a shape.
Triangles
The Interior Angles of a Triangle add up to 180°
90° + 60° + 30° = 180°
80° + 70° + 30° = 180°
It works for this triangle!
Let's tilt a line by 10° ...
It still works, because one angle went up by
10°, but the other went down by 10°
9 Angles & Shapes
Page 1
Name of Lecturer: Mr. J.Agius
Course: HVAC1
Quadrilaterals (Squares, etc)
(A Quadrilateral is any shape with 4 sides)
90° + 90° + 90° + 90° = 360° 80° + 100° + 90° + 90° = 360°
A Square adds up to 360°
Let's tilt a line by 10° ... still adds up to 360°!
The Interior Angles of a Quadrilateral add up to 360°
Because there are Two Triangles in a Square
The internal angles
in this triangle add
up to 180°
(90°+45°+45°=180°)
... and for this square
they add up to 360°
... because the
square can be made
from two triangles!
Pentagon
A pentagon has 5 sides, and can be made from three triangles, so
you know what ...
... its internal angles add up to 3 × 180° = 540°
And if it is a regular pentagon (all angles the same), then each
angle is 540° / 5 = 108°
(Exercise: make sure each triangle here adds up to 180°, and
check that the pentagon's internal angles add up to 540°)
9 Angles & Shapes
Page 2
Name of Lecturer: Mr. J.Agius
Course: HVAC1
The General Rule
So, each time we add a side (triangle to quadrilateral, quadrilateral to
pentagon, etc), we add another 180° to the total:
(Note: it is a Regular Polygon when all sides are equal, all angles are equal.)
If it is a Regular Polygon...
Shape
Sides
Sum of
Internal Angles
Triangle
3
180°
60°
Quadrilateral
4
360°
90°
Pentagon
5
540°
108°
Hexagon
6
720°
120°
Heptagon (or
Septagon)
7
900°
128.57...°
Octagon
8
1080°
135°
...
...
..
Any Polygon
n
(n-2) × 180°
Shape
Each Angle
...
...
(n-2) × 180° / n
That last line can be a bit hard to understand, so let's have one example:
Example: What about a Regular Decagon (10 sides)?
Sum of Internal Angles == (n-2) × 180°
== (10-2)×180° = 8×180°
== 1440°
And it is a Regular Decagon so:
Each internal angle = 1440°/10 = 144°
These notes were taken from http://www.mathsisfun.com/geometry/interior-anglespolygons.html
Go to this website to practice more.
For more info go to: http://www.mathopenref.com/polygoninteriorangles.html
9 Angles & Shapes
Page 3
Name of Lecturer: Mr. J.Agius
Course: HVAC1
Exercise 1:
Find the size of each interior angle of a regular polygon with
a)
5 sides
b)
20 sides
c)
24 sides
d)
36 sides
Find the number of sides of a regular polygon with an interior angle of
a)
140°
b)
160°
c)
162°
d)
170°
Exercise 2:
In each of the following diagrams find the marked angles.
1.
2.
o
xo
x
130o
165o
63o
100o
160o
137o
145o
3.
4.
4w
o
3wo
xo
4w
85o
o
4wo
60o
3wo
9 Angles & Shapes
xo
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Name of Lecturer: Mr. J.Agius
Course: HVAC1
Exercise 3:
In the following questions, find the value of x.
1.
2.
2xo
3xo
xo
100o
3xo
130o
xo
xo
100o
xo
3.
2xo
4.
2xo
4xo
xo
xo
xo
2xo
2xo
2xo
60o
o
3x
Exercise 4:
1.
ABCDE is a regular pentagon. BC and ED are produced (i.e. extended) and
meet in F. Find each of the angles in triangle CDF.
C
B
A
F
D
E
9 Angles & Shapes
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Name of Lecturer: Mr. J.Agius
2.
a)
Find
Course: HVAC1
i) an exterior angle,
ii) interior angle, for a regular hexagon.
b)
ABCDEF is a regular hexagon. Find the size of
i) FAE
ii) EAB
iii) BEF
iv) AEB
A
B
F
C
E
3.
D
The diagram shows a hexagon with just two lines of symmetry. These are
marked PQ and RS. Find the values of x, y and z.
P
R
65o
xo
yo
S
zo
Q
9 Angles & Shapes
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Name of Lecturer: Mr. J.Agius
4.
Course: HVAC1
ABCDEFGH is a regular octagon and O is equidistant from all the vertices.
Find the angles in triangle AOB.
A
B
H
C
O
D
G
Exercise 5:
F
E
Solve the following questions.
1.
How many sides does a polygon have if each angle is 162°?
2.
What is a polygon called if each interior angle equals 60°?
3.
What is the measure of each interior angle in a regular octagon?
4.
What is the measure of a central angle if each interior angle of the polygon
is 108°?
5.
If the sum of the interior angles of a polygon equals 3600°, how many
degrees in each interior angle does the polygon have?
6.
How many degrees in each interior angle does a polygon have if the sum
of its interior angles is 1980°?
7.
What is the measure of each interior angle if the sum of its interior angles
is 1620°?
8.
What is the measure of each interior angle in a regular fifteen sided
polygon?
9.
If the measure of a central angle in a regular polygon is 45, what is the
measure of each interior angle of the given polygon?
10.
What is the size of each interior angle of a twenty-one sided regular
polygon?
9 Angles & Shapes
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