Lesson 9.1 · 25 min

Angles in Polygons

The interior angle sum $180 (n - 2)$ , the exterior angle sum of $36 0^{\circ}$ , and the angles of regular polygons.

In this lesson you will learn to…

Explain why the interior angles of an $n$ -sided polygon add up to $18 0^{\circ} (n - 2)$
Find a missing interior angle in any polygon
Know that the exterior angles of any polygon add up to $36 0^{\circ}$
Find the interior and exterior angles of a regular polygon, and its number of sides

Why this matters

Architects, tile designers and engineers constantly need the angles of polygons — to cut a frame for a hexagonal window, lay floor tiles, or design a nut that a spanner can grip. One formula covers every polygon there is.

Why $180(n - 2)$ ?

Shape	Sides, $n$	Triangles, $n - 2$	Interior angle sum
Triangle	3	1	$180^\circ$
Quadrilateral	4	2	$360^\circ$
Pentagon	5	3	$540^\circ$
Hexagon	6	4	$720^\circ$
Octagon	8	6	$1080^\circ$
Decagon	10	8	$1440^\circ$

A second way to see it: join the centre to every vertex to make $n$ triangles ( $180n$ in total), then subtract the full turn of $360^\circ$ at the centre: $180n - 360 = 180(n - 2)$ .

Nadia’s mistake. “A 12-sided shape has 3 times as many sides as a quadrilateral, so its angle sum is $3 \times 360^\circ$ .” No — it is $180 \times 10 = 1800^\circ$ , not $1080^\circ$ . The number of triangles does not triple.

Why the exterior angles make $360^\circ$

Walk round the outside of any polygon. At each corner you turn through the exterior angle. By the time you are back where you started, facing the same way, you have turned through exactly one full turn: $360^\circ$ .

Is it a polygon at all?

The interior angle sum is always a multiple of $180^\circ$ . That is why $4180^\circ$ cannot be the angle sum of any polygon: $4180 \div 180$ is not a whole number.

Key rules and formulas

Interior angle sum
$sum of interior angles = 18 0^{\circ} \times (n - 2)$
From one vertex you can split an $n$ -sided polygon into $n - 2$ triangles, each worth $18 0^{\circ}$ .
Interior + exterior = 180°
At every vertex the interior and exterior angles lie on a straight line, so they add to $18 0^{\circ}$ .
Exterior angle sum
$sum of exterior angles = 36 0^{\circ}$
True for every polygon. Walking all the way round the shape, you turn through one full turn.
Regular polygons
$exterior angle = \frac{36 0 ^{\circ}}{n} interior angle = 18 0^{\circ} - exterior$
All the angles are equal. Knowing either angle lets you find $n$ : exterior $1 8^{\circ}$ gives $n = 360 \div 18 = 20$ sides.

Try it

Polygon angle explorer

Number of sides5

Name: pentagon
Interior angle sum = (n − 2) × 180°: (5 − 2) × 180° = 540°
Each interior angle: 108°
Each exterior angle = 360° ÷ n: 72°

Change the number of sides and watch the angle sum go up by $18 0^{\circ}$ each time — one more triangle.

Worked examples

Worked example 1

Interior angle sum of a hexagon

Find the sum of the interior angles of a hexagon.

Try it yourself first, then reveal the solution one step at a time.

Worked example 2

A missing angle

A pentagon has angles

9 0^{\circ}

9 0^{\circ}

11 0^{\circ}

13 0^{\circ}

and

a

. Find

a

Try it yourself first, then reveal the solution one step at a time.

Worked example 3

How many sides?

A regular polygon has interior angles of

15 0^{\circ}

. How many sides does it have?

Try it yourself first, then reveal the solution one step at a time.

Practice: angles in polygons

10 questions

Ten questions. Angles are in degrees — type the number only.

Question 1 of 10Complete the tableFoundation

Complete the table.

Shape	Sides, n	n − 2	Sum of interior angles (°)
Pentagon	Row 1, Sides, n	Row 1, n − 2	Row 1, Sum of interior angles (°)
Hexagon	Row 2, Sides, n	Row 2, n − 2	Row 2, Sum of interior angles (°)
Heptagon	7	Row 3, n − 2	Row 3, Sum of interior angles (°)
Octagon	Row 4, Sides, n	Row 4, n − 2	Row 4, Sum of interior angles (°)
Decagon	10	Row 5, n − 2	Row 5, Sum of interior angles (°)

Question 2 of 10Type the answer
A polygon has an interior angle sum of $270 0^{\circ}$ . How many sides does it have?
Your answer
Question 3 of 10Type the answer
A hexagon has angles $3 8^{\circ}$ , $17 0^{\circ}$ , $11 8^{\circ}$ , $10 3^{\circ}$ , $16 5^{\circ}$ and $b$ . Find $b$ .
Your answer°
Question 4 of 10Type the answer
A pentagon has angles $14 1^{\circ}$ , $10 9^{\circ}$ , $11 0^{\circ}$ , $d$ and $d$ . Find $d$ .
Your answer°
Question 5 of 10Explore and match
Find the regular polygon whose interior angle is $14 4^{\circ}$ .
Move the controls until you match the question, then press Check.
Number of sides5
Name
pentagon
Interior angle sum = (n − 2) × 180°
(5 − 2) × 180° = 540°
Each interior angle
108°
Each exterior angle = 360° ÷ n
72°

Question 6 of 10Complete the table

Complete the angles of these regular polygons.

Regular polygon	Exterior angle (°)	Interior angle (°)
Pentagon	Row 1, Exterior angle (°)	Row 1, Interior angle (°)
Hexagon	Row 2, Exterior angle (°)	Row 2, Interior angle (°)
Octagon	Row 3, Exterior angle (°)	Row 3, Interior angle (°)
Nonagon	Row 4, Exterior angle (°)	Row 4, Interior angle (°)

Question 7 of 10Type the answer
A regular polygon has an exterior angle of $1 8^{\circ}$ . How many sides does it have?
Your answer
Question 8 of 10Multiple choice
Nadia says a 12-sided polygon has an angle sum of $3 \times 36 0^{\circ} = 108 0^{\circ}$ . What is the correct sum?
Question 9 of 10True or false
$418 0^{\circ}$ could be the sum of the interior angles of a polygon.
Question 10 of 10Step by stepChallenge
Use ratio and the angle sum.
The six interior angles of a hexagon are in the ratio $4 : 5 : 6 : 6 : 7 : 8$ . Find the largest angle.
1. What is the total of the angles in a hexagon?
  Step 1 answer°

In this lesson you will learn to…

Why this matters

Why 180(n−2)180(n - 2)180(n−2)?

Why the exterior angles make 360∘360^\circ360∘

Is it a polygon at all?

Key rules and formulas

Interior angle sum

Interior + exterior = 180°

Exterior angle sum

Regular polygons

Polygon angle explorer

Worked examples

Practice: angles in polygons

Why $180(n - 2)$ ?

Why the exterior angles make $360^\circ$