KS3 Maths · Year 8 · Chapter 3: Polygons · Lesson 2

Interior and exterior angles of regular polygons

Year 8About 60 minutesNo calculator48 marks

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Date

Do Now

Quick questions to start.

A1
Needed today
Work out $360 \div 24$ .
[1 mark]
Answer
A2
Last lesson
Work out the sum of the interior angles of an octagon.
[1 mark]
Answer
A3
Year 7
Two angles on a straight line. One is $14 8^{\circ}$ . Work out the other.
[1 mark]
Answer
A4
From chapter 1
Write $54$ as a percentage of $360$ .
[1 mark]
Answer

Example, then your turn

Year 8

Your teacher works through each example with you. Then try the one beside it.

Example 1

A regular dodecagon has $12$ equal sides and $12$ equal angles. Work out the size of each exterior angle and each interior angle.

Hint The exterior angles add up to

36 0^{\circ}

and they are all equal. Work out the exterior angle first.

B1
A regular polygon has $15$ sides. Work out the size of each exterior angle and each interior angle.
[2 marks]
Answerexterior $\dots \dots^{\circ}$ interior $\dots \dots^{\circ}$

Example 2

Each exterior angle of a regular polygon is $2 0^{\circ}$ . How many sides does it have?

Hint The exterior angles add up to

36 0^{\circ}

. How many lots of

2 0^{\circ}

make

36 0^{\circ}

B2
Each exterior angle of a regular polygon is $1 5^{\circ}$ . How many sides does it have?
[2 marks]
Answer $\dots\dots$ sides

Example 3

Each interior angle of a regular polygon is $17 0^{\circ}$ . How many sides does it have?

Hint Work out the exterior angle first.

B3
Each interior angle of a regular polygon is $17 2^{\circ}$ . How many sides does it have?
[2 marks]
Answer $\dots\dots$ sides

Example 4

$A B C D E F$ is a regular hexagon. Work out the size of angle $B A C$ .

Not drawn accurately

Hint

A B = B C

, so triangle

A B C

is isosceles. Find angle

A B C

first.

B4
$A B C D E F G H$ is a regular octagon. Work out the size of angle $B A C$ .
[3 marks]
Not drawn accurately
Answerangle $B A C = \dots \dots^{\circ}$

Practice

Year 8

C1
Could a regular polygon have an exterior angle of $2 5^{\circ}$ ? Give a reason for your answer.
[2 marks]
C2
A regular polygon has $14$ sides. Work out the size of each exterior angle. Give your answer as a mixed number.
[2 marks]
Answer $\dots \dots^{\circ}$

C3
Each interior angle of a regular polygon is $17 4^{\circ}$ . Work out the sum of its interior angles.
[3 marks]
Answer $\dots \dots^{\circ}$
C4
$O$ is the centre of this regular decagon. Work out the size of angle $A O B$ .
[2 marks]
Not drawn accurately
Answerangle $A O B = \dots \dots^{\circ}$
C5
A regular polygon has $40$ sides. Work out the size of each interior angle.
[2 marks]
Answer
C6
The interior angle of a regular polygon is $5 4^{\circ}$ more than the interior angle of a regular pentagon. How many sides does the polygon have?
[2 marks]
Answer $\dots\dots$ sides

Exam-style questions

GCSE Foundation

D1
Mia says, "I can draw a regular polygon with an interior angle of $13 0^{\circ}$ ."
Explain why Mia is wrong.
[2 marks]

D2
$A B C D E F$ is a regular hexagon and $A B G H$ is a square. $F$ is joined to $H$ .
Not drawn accurately
(a)
Work out the size of angle $F A H$ .
[2 marks]
Answer
(b)
Work out the size of angle $A H F$ . Give a reason for each step.
[3 marks]
Answerangle $A H F = \dots \dots^{\circ}$
Total for D2: 5 marks
D3
The interior angle of a regular polygon is $15 6^{\circ}$ more than its exterior angle. How many sides does the polygon have? You must show your working.
[4 marks]
Answer $\dots\dots$ sides
D4
In a regular polygon, interior angle : exterior angle $= 7 : 2$ . How many sides does the polygon have?
[4 marks]
Answer $\dots\dots$ sides
D5
Each interior angle of a regular polygon is $x^{\circ}$ and each exterior angle is $(x - 140)^{\circ}$ . Work out the value of $x$ , then the number of sides.
[3 marks]
Answer $x = \dots\dots$ $\dots\dots$ sides

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
Some regular polygons have angles that are whole numbers of degrees: a regular hexagon has exterior angle $6 0^{\circ}$ and interior angle $12 0^{\circ}$ . How many different regular polygons have exterior angles that are whole numbers of degrees?
[4 marks]
Hint 1 · What to try
The exterior angle is $360 \div n$ . For it to be a whole number, $n$ must divide $360$ exactly.
Hint 2 · The first line
List all the factors of $360$ . There are $24$ of them.
Hint 3 · The full method
A polygon has at least $3$ sides, so remove the factors $1$ and $2$ : $24 - 2 = 22$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can work out the exterior and interior angles of a regular polygon.
I can find the number of sides of a regular polygon from one of its angles.
I can find angles in diagrams made of regular polygons.

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