Download Yr8-InteriorExteriorAnglesWorksheet

Year 8 Interior and Exterior Angles
Key Formulae for Polygons:
2) The total of the interior angles of a polygon
is 1260°. How many sides does it have?
(in terms of number of sides 𝑛):
Total exterior angle:
Total interior angle:
Exterior angle of
regular polygon:
Interior angle of
regular polygon:
________________________
1) The interior angle of a regular polygon is
179°. How many sides does it have?
________________________
Exercise 1
2) If a 𝑛-sided polygon has exactly 3 obtuse
angles (i.e. 90° < 𝜃 < 180°), then determine
the possible values of 𝑛 (Hint: determine the
possible range for the sum of the interior
angles, and use these inequalities to solve).
1) Determine the angle 𝑥.
b)
a)
105°
100°
𝑥
𝑥
50°
c)
d)
___________________
Exterior vs Interior Angles
An exterior angle is ____________________
e)
48°
f)
50°
𝑥
____________________________________
94°
69°
260°
92°
113°
g)
𝑥
110°
300°
43°
h)
𝑥
95°
160°
𝑥
61°
𝑥
i)
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At a vertex, 𝐼𝑛𝑡𝑒𝑟𝑖𝑜𝑟 𝑎𝑛𝑔𝑙𝑒 +
𝑒𝑥𝑡𝑒𝑟𝑖𝑜𝑟 𝑎𝑛𝑔𝑙𝑒 = ______
Num
Sides
3
4
5
6
7
8
9
10
Name
Triangle
Quadrilateral
Pentagon
Hexagon
Heptagon
Octagon
Nonagon
Decagon
Exterior
Angle
Interior
Angle
4. [GCSE]
Examples:
1)
2)
3) What is the exterior angle of a 180-sided
regular polygon?
__________________________________
4) The interior angle of a regular polygon is
165. How many sides does it have?
__________________________________
Exercise 2
1. Determine how many sides a regular
polygon with the following exterior angle
would have:
a. 30° ______________
b. 45° ______________
c. 12° ______________
d. 9° ______________
2. Determine how many sides a regular
polygon with the following interior angle
would have:
a. 156° ______________
b. 162° ______________
c. 144° ______________
d. 175° ______________
3. [GCSE]
The diagram shows a regular hexagon and
a regular octagon. Calculate the size of
the angle marked 𝑥. You must show all
your working.
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The pattern is made from two types of
tiles, tile A and tile B.
Both tile A and tile B are regular polygons.
Work out the number of sides tile A has.
_________________________________
5. A regular polygon is surrounded by
squares and regular hexagons, alternating
between the two. How many sides does
this shape have?
_________________________________
6. Find all regular polygons which tessellate
(when restricted only to one type of
polygon).
__________________________________
1. By thinking about interior angles, prove
that the regular polygons you identified above
are the only regular polygons which tessellate.
Answers:
Exercise 1
Q1
a)
b)
c)
d)
e)
f)
g)
h)
i)
75°
25°
222°
309°
54°
120°
252°
2) 𝟏𝟖𝟎(𝒏 − 𝟐) = 𝟏𝟐𝟔𝟎 so 𝒏 = 𝟗
1)
𝟏𝟖𝟎(𝒏 − 𝟐)
= 𝟏𝟕𝟗
𝒏
𝟏𝟖𝟎(𝒏 − 𝟐) = 𝟏𝟕𝟗𝒏
𝟏𝟖𝟎𝒏 − 𝟑𝟔𝟎 = 𝟏𝟕𝟗𝒏
𝒏 = 𝟑𝟔𝟎
2) If 3 angles are obtuse, the sum of these,
say 𝑂, has the range 270 < 𝑂 < 540. For the
𝑛 − 3 angles that are not obtuse (i.e. acute or
right-angled), then the sum 𝐴 has the range:
0 < 𝐴 ≤ 90(𝑛 − 3). The total of the interior
angles is 180(𝑛 − 2), so
270 < 180(𝑛 − 2) < 540 + 90(𝑛 − 3)
Solving 270 < 180(𝑛 − 2), we get 𝑛 > 3.5
and solving 180(𝑛 − 2) < 540 + 90(𝑛 − 3),
we get 𝑛 < 7. Thus 𝑛 = 4, 5 𝑜𝑟 6.
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