1MA1 · G3–G4Geometry & MeasuresFoundation & HigherGrades 3–6

Polygons

Q: Can a regular polygon have an exterior angle of 50°?

No. 360 50 = 7.2 is not a whole number.

A polygon is a closed shape with straight sides — triangles, pentagons, octagons. Two numbers unlock almost every polygon angle question: the interior angles of an $n$ -sided polygon always add up to $(n - 2) \times 180^{\circ}$ , and the exterior angles of any polygon always add up to $360^{\circ}$ .

40 min Angles & Geometric Reasoning

The big picture

Both facts come from simple arguments. Split a polygon into triangles from one corner and you get $n - 2$ triangles, each contributing $180^{\circ}$ . Walk around the outside of a polygon and you turn through exactly one full turn, $360^{\circ}$ , however many sides it has.

For regular polygons, the exterior angle $\dfrac{360^{\circ}}{n}$ is the quickest route to everything else — including working out how many sides a polygon has from one of its angles, a regular Edexcel question.

What you'll learn

Find the sum of interior angles of any polygon
Use the sum of exterior angles to find angles in regular polygons
Find the number of sides of a regular polygon from an angle
Recall the properties of special quadrilaterals

Interior angle sum

From one vertex, draw diagonals to split an $n$ -sided polygon into $n - 2$ triangles. Each triangle has angles summing to $180^{\circ}$ .

So the interior angles of an $n$ -gon sum to $(n - 2) \times 180^{\circ}$ : a pentagon $540^{\circ}$ , a hexagon $720^{\circ}$ , a decagon $1440^{\circ}$ .

In a $\textbf{regular}$ polygon, all sides and angles are equal, so each interior angle is the sum divided by $n$ .

S = (n - 2) \times 180^{\circ}

Sum of interior angles.

Example 1Find the size of each interior angle of a regular decagon.

1Sum:

(10 - 2) \times 180^{\circ} = 1440^{\circ}

2Each angle:

1440^{\circ} \div 10

Answer

144^{\circ}

Your turnSame method, new numbers — try it before moving on.

Find the size of each interior angle of a regular 18-sided polygon.

Your answer

Show worked solution

Solution

1Exterior angle $= 360 \div 18 = 20^\circ$ .
2Interior angle $= 180 - 20 = 160^\circ$ .

Answer: $160^\circ$

Exterior angles

An exterior angle is formed by extending one side; it and the interior angle at that vertex lie on a straight line, so they sum to $180^{\circ}$ .

The exterior angles of any polygon sum to $360^{\circ}$ .

For a regular polygon, each exterior angle is $\dfrac{360^{\circ}}{n}$ , so $n = \dfrac{360^{\circ}}{\text{exterior angle}}$ .

\text{exterior} = \dfrac{360^{\circ}}{n}

Regular polygon.

\text{interior} + \text{exterior} = 180^{\circ}

At each vertex.

Example 1Each exterior angle of a regular polygon is

24^{\circ}

. How many sides does it have?

n = 360 \div 24

Answer15 sides

Your turnSame method, new numbers — try it before moving on.

Each exterior angle of a regular polygon is $20^\circ$ . How many sides does it have?

Your answer

Show worked solution

Solution

1 $360 \div 20 = 18$ sides.

Answer: $18$

Why this works

The exterior angle route avoids the interior angle formula entirely. To find the number of sides from an interior angle of $156^{\circ}$ , take the exterior angle $24^{\circ}$ and divide it into $360^{\circ}$ .

Special quadrilaterals

$\textbf{Square}$ : four equal sides, four right angles, diagonals equal and bisect at right angles.

$\textbf{Rectangle}$ : opposite sides equal, four right angles, diagonals equal and bisect each other.

$\textbf{Parallelogram}$ : opposite sides parallel and equal, opposite angles equal, diagonals bisect each other.

$\textbf{Rhombus}$ : four equal sides, opposite angles equal, diagonals bisect at right angles. $\textbf{Kite}$ : two pairs of adjacent equal sides, one pair of equal opposite angles, diagonals perpendicular. $\textbf{Trapezium}$ : one pair of parallel sides.

Your turnSame method, new numbers — try it before moving on.

One angle of a rhombus is $64^\circ$ . Find the size of its obtuse angles.

Your answer

Show worked solution

Solution

1Adjacent angles of a rhombus (a parallelogram) sum to $180^\circ$ .
2 $180 - 64 = 116^\circ$ .

Answer: $116^\circ$

Tip — A square is a special rectangle and a special rhombus. When asked to "name" a shape, give the most specific name the properties allow.

Polygon problems

Combine polygon facts with other angle facts: tessellations, shapes meeting at a point, or polygons joined along an edge.

Shapes tessellate around a point only if their angles there sum to $360^{\circ}$ .

Example 1A regular hexagon and a regular octagon share one side and meet a third regular polygon at a point, with no gaps. How many sides does the third polygon have?

1Hexagon interior angle:

720 \div 6 = 120^{\circ}

. Octagon:

1080 \div 8 = 135^{\circ}

2Remaining angle at the point:

360 - 120 - 135 = 105^{\circ}

3Its exterior angle would be

75^{\circ}

, and

360 \div 75 = 4.8

AnswerNot a whole number — so no regular polygon fits. The gap of

105^{\circ}

cannot be filled by a regular polygon.

Think like an examiner

Show the interior sum calculation before dividing.
Use exterior angles to find the number of sides.
State the properties used when classifying quadrilaterals.

Polygons

(n - 2) \times 180^{\circ}

Interior angle sum.

360^{\circ}

Exterior angle sum.

n = \tfrac{360^{\circ}}{\text{exterior}}

Sides of a regular polygon.

Watch out for these

Using

n \times 180^{\circ}

for the interior sum.

It is

(n - 2) \times 180^{\circ}

Dividing 360 by the interior angle to find

n

Divide 360 by the exterior angle.

Applying regular-polygon formulas to irregular polygons.

Equal angles only in regular polygons; sums work for all.

Saying a rhombus has four right angles.

Only if it is a square.

Stretch yourself

In a regular polygon, each interior angle is four times the size of each exterior angle. How many sides does the polygon have, and what is the sum of its interior angles?

Hint — Let the exterior angle be $x$ . Interior and exterior at a vertex sum to $180^{\circ}$ .

Reveal solution

SolutionWith exterior angle

x

, the interior angle is

4x

, and

x + 4x = 180^{\circ}

, so

x = 36^{\circ}

. Then

n = 360 \div 36 = 10

sides. The interior sum is

(10 - 2) \times 180^{\circ} = 1440^{\circ}

. Check: each interior angle is

144^{\circ} = 4 \times 36^{\circ}

✓.

\blacksquare

Questions students ask

Why do exterior angles always sum to 360°?

Walking around the polygon, you turn by each exterior angle and end facing the way you started — one full turn.

Can a regular polygon have an exterior angle of 50°?

No.

360 \div 50 = 7.2

is not a whole number.

Is a square a rhombus?

Yes — it has four equal sides. It is also a rectangle.

Key takeaways

Interior angle sum of an $n$ -gon: $(n - 2) \times 180^{\circ}$ .
Exterior angles of any polygon sum to $360^{\circ}$ .
Regular polygon: exterior $= 360^{\circ}/n$ ; interior $= 180^{\circ} -$ exterior.
Find $n$ by dividing $360^{\circ}$ by the exterior angle.
Know the side, angle and diagonal properties of special quadrilaterals.

Exam-style questions

Practise it the way it is examined

3 original questions · 8 marks, written to match the style and mark allocation of the real papers. Work on paper, then open the mark scheme and tick the marks you earned — method marks count even if the final answer slips.

Question 1Grade 32 marks

Work out the sum of the interior angles of a 9-sided polygon.[2]
Show mark scheme
- M1 $(9 - 2) \times 180$
- A1 $1260^\circ$
Answer: $1260^\circ$

Question 2Grade 53 marks

The angles of a pentagon are $x$ , $x + 5$ , $x + 10$ , $2x$ and $2x$ degrees.

Work out the size of the largest angle.[3]
Show mark scheme
- M1Angle sum $540^\circ$
- M1 $7x + 15 = 540$
- A1 $x = 75$ , largest angle $150^\circ$
Answer: $150^\circ$

Question 3Grade 63 marks

Each interior angle of a regular polygon is 5 times its exterior angle. How many sides does it have?[3]
Show mark scheme
- M1 $e + 5e = 180$
- M1 $e = 30$ , $360 \div 30$
- A1 $12$
Answer: $12$

Unlimited practice

Skill drill

A new question every time, marked instantly, with a full worked solution. Aim for a streak of five, then move up a level.

0 / 0 first-time correct Streak 0Best at this level: 0

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