Shape and Space - part 2
REGULAR POLYGONS A
The word ‘polygon’ means: with many angles. A shape is usually called
regular if all its angles and all its sides, are equal. These are some of
the most common regular polygons:
Triangle
Square
Pentagon
Hexagon
Heptagon
Octagon
External angles
e
The outside, or external, angles
of a shape add up to 360° no
matter how many sides it has.
e
e
(If you walked around this shape,
you would make a turn of 360°.)
e
e
e + e + e + e + e = 360°
Example:
e
e
e
So if you want to find the external angle of a regular polygon, you can
divide 360° by the number of sides.
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In this regular equilateral triangle:
360°  3 = 120°
The external angle (e) = 120°
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Shape and Space - part 2
A)
Try using this method to find the external (outside) angle of
these polygons
1)
3)
5)
7)
Octagon (8 sides)
Hexagon (6 sides)
A 12 sided figure
A 30 sided figure
2)
4)
6)
Pentagon (5 sides)
Decagon (10 sides)
A 15 sided figure
Internal angles
You can use this method to work out the inside or internal angle too.
The external and internal angles lie next to each other on a straight
line, and so add up to 180°.
Once you know the external angle, you can subtract it from 180° to
find the internal angle.
Example:
The external angle of this triangle equals
360°  3
= 120°
The internal angle (a) equals
180° - 120°
= 60°
B)
a
a
Try to work out the internal angle of each shape in the previous
questions 1 - 7
Remember: internal < = 180° - external <
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a
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Shape and Space - part 2
Working out the number of sides
If you know the internal or external angle of a polygon, you can work
back to find out how many sides it has:
Just see how many times the external angle divides into 360°.
Example:
How many sides has a regular shape with an internal angle of 135?
First find the external angle:
External angle
= 180° - 135°
= 45°
Number of sides = 360°  45°
=8
135
C)
Try these questions
1)
How many sides has a regular shape with an internal angle of
162°?
2)
How many sides has a regular shape with an internal angle of
140°?
f lex i p
Check your answers then discuss with your tutor what you need to
work on next.
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Shape and Space - part 2
Answers
Regular Polygons A
A)
External Angles
1)
45°
2)
72°
3)
60°
4)
36°
5)
30°
6)
24°
7)
12°
B)
Internal Angles
1)
135°
2)
108°
3)
120°
4)
144°
5)
150°
6)
156°
7)
168°
C)
20
2)
9
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1)
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Shape and Space - part 2
REGULAR POLYGONS B
To find the angle sum of a regular polygon:
The shapes below can be split into a number of triangles.
Try to complete the table ( the triangles shouldn't overlap).
Triangle
Square
Number
of sides
3
4
Number
of triangles
1
2
Pentagon
Hexagon
Heptagon
Octagon
Number
1 x 180 2 x 180
of degrees
(angle sum) = 180
= 360
1)
Can you use your investigation to work out the rule for finding
the angle sum of any shape?
(Check the answer page if you are not sure)
What will be the angle sum of a shape with 9 sides?
3)
What is the total number of degrees in a shape with 11 sides?
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2)
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Shape and Space - part 2
Can you think of a way to reverse this and find the number of sides
from the total number of degrees?
Example:
How many sides are there in a shape with angles totalling 1980?
Firstly, how many triangles are there in this shape?
1980 ÷ 180 = 11 triangles
Total number of sides = 11 + 2
= 13 sides
4)
How many sides will there be in a shape with an angle sum of
2520?
f lex i p
Check your answers then discuss with your tutor what you need to
work on next.
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Shape and Space - part 2
Answers
Regular Polygons B
Number
of sides
3
4
5
6
7
8
Number
of triangles
1
2
3
4
5
6
Number
of degrees
180
360
540
720
900
1080
1)
The number of triangles = the number of sides minus 2.
To find the angle sum: multiply the number of sides minus 2 by 180
9-2=7
7 x 180 = 1260
3)
11 - 2 = 9
9 x 180 = 1620
4)
2520 ÷ 180 = 14 triangles
14 + 2 = 16 sides
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2)
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