GCSE Interior and Exterior Angles
2) If a 𝑛-sided polygon has exactly 3 obtuse
angles (i.e. 90° < 𝜃 < 180°), the rest acute or
right-angled, 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).
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Exercise 1
1) Determine the angle 𝑥.
b)
a)
105°
100°
𝑥
𝑥
50°
c)
(Lesson 2) Test Your Understanding:
d)
e)
48°
f)
50°
𝑥
94°
69°
260°
92°
113°
g)
𝑥
1) Determine the exterior angle of a 40sided regular polygon.
2) Determine the interior angle of a 15sided regular polygon.
3) The interior angle of a regular polygon
is 178°. Determine how many sides it
has.
4)
110°
300°
43°
h)
𝑥
95°
160°
𝑥
5) [Edexcel] The diagram shows a regular
hexagon and a regular octagon.
Calculate the size of the angle marked
𝑥. You must show all your working.
61°
𝑥
i)
Exercise 2
2) The total of the interior angles of a polygon
is 1260°. How many sides does it have?
________________________
1) The interior angle of a regular polygon is
179°. How many sides does it have?
________________________
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1) [Edexcel IGCSE Jan2014(R)-3H
Q3b] The diagram shows a
regular 6-sided polygon.
Work out the value of 𝑦.
2) [Edexcel IGCSE Jan2014(R)-3H
Q3a] The diagram shows a
regular 5-sided polygon. Work
out the value of 𝑥.
3) Determine the value of 𝑎.
4) Determine how many sides a regular
polygon with the following exterior angle
would have:
a. 30° ______________
b. 45° ______________
c. 12° ______________
d. 9° ______________
5) Determine how many sides a regular
polygon with the following interior angle
would have:
a. 156° ______________
b. 162° ______________
c. 144° ______________
d. 175° ______________
6) [Edexcel GCSE
Mar13-1H Q13]
The diagram shows
a square and 4
regular pentagons.
Work out the size
of the angle marked 𝑥.
7) [Edexcel IGCSE
May2015(R)-3H
Q2b]
The diagram
shows 3 identical
regular
pentagons. Work
out the value of 𝑦.
8) [Edexcel IGCSE
May2015(R)-3H Q2b]
The diagram shows 3
identical regular
pentagons. Work out
the value of 𝑦.
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9) [JMO 2007 A6] The sizes in degrees of the
interior angles of a pentagon are
consecutive whole numbers. What is the
size of the largest of these angles?
10) [IMC 2015 Q9] What
is the value of 𝑝 +
𝑞+𝑟+𝑠+𝑡+𝑢+
𝑣 + 𝑤 + 𝑥 + 𝑦 in the
diagram?
____________________________________
Lesson 3: Test Your Understanding 1
The diagram shows 4 congruent
regular pentagons that form the
sides of an 𝑛-sided regular polygon.
Determine the value of 𝑛.
Test Your Understanding 2
[IMC 2006 Q19] The diagram
shows a regular pentagon and a
regular hexagon which overlap.
What is the value of 𝑥?
Test Your Understanding 3
[IMC 2009 Q12] The diagram
shows a square inside a regular
hexagon. What is the size of the
marked angle at 𝑋?
Exercise 3
1) [KS3 SATs 2004 L6-L8 Paper 2
Q19 Edited] A pupil has three
tiles. One is a regular octagon,
one is a regular hexagon, and
one is a square. The side
length of each tile is the same.
The pupil says the hexagon will fit exactly
like this. Is the pupil correct?
2) [Edexcel IGCSE
Nov2009-3H Q3a]
The diagram shows a
regular octagon,
with centre O. Work
out the value of 𝑥.
7) [IMC 2003 Q22] The
diagram shows a regular
dodecagon (a polygon with
twelve equal sides and
equal angles). What is the
size of the marked angle?
3) [Edexcel IGCSE Nov-2010-4H Q13]
The size of each interior angle of a regular
polygon is 11 times the size of each
exterior angle. Work out the number of
sides the polygon has.
8) [JMO 2014 B1] The figure
shows an equilateral
triangle ABC, a
square BCDE, and a regular
pentagon BEFGH.
What is the difference
between the sizes
of ∠ADE and ∠AHE?
4) [Edexcel GCSE Nov2014-1H Q17]
ABCDEFGH is a regular octagon. BCKFGJ is
a hexagon. JK is a line of symmetry of the
hexagon. Angle 𝐵𝐽𝐺 = angle 𝐶𝐾𝐹 =
140°. Work out the size of angle KFE.
5) A regular polygon
𝐴 is surrounded by
squares and
equilateral
triangles in an
alternating
pattern, as shown. Show that 𝐴 is a
hexagon.
6) A regular
polygon 𝐵 with
𝑛 sides is
surrounded by
squares and
regular
pentagons in an alternating pattern, as
shown. Determine the value of 𝑛.
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9) [IMC 2005 Q14] Ten stones, of identical
shape and size, are used to make an arch,
as shown in the diagram. Each stone has a
cross-section in the shape of a trapezium
with three equal sides. What is the size of
the smallest angles of the trapezium?
10) [IMC 2018 Q18] The diagram
shows a regular pentagon and
an equilateral triangle placed
inside a square. What is the
value of 𝑥?
11) 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.