Interior and Exterior Angles of Polygons

Use the angle sum (n − 2) × 180°, exterior angles that add to 360° and interior + exterior = 180° to find angles and the number of sides of a polygon, even when two shapes meet.

In the exam for Cambridge IGCSE Maths 0580, Edexcel IGCSE Maths 4MA1, Cambridge O Level Maths 4024 · 98 real past-paper questions below

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Three Angle Facts for Polygons

A polygon is a flat shape with straight sides. From one corner you can cut it into triangles: always two fewer triangles than sides.

The sum of the interior angles is:

Formula

S = (n - 2) \times 18 0^{\circ}

$n$ is the number of sides. A pentagon has $3 \times 18 0^{\circ} = 54 0^{\circ}$ .

Fact	Value
Exterior angles, all added	always $36 0^{\circ}$
Interior + exterior at a corner	$18 0^{\circ}$ (a straight line)
Regular polygon	all sides and all angles equal

Key idea

Regular octagon: each exterior angle is $36 0^{\circ} \div 8 = 4 5^{\circ}$ , so each interior angle is $13 5^{\circ}$ .

Key idea

Why do bees build hexagons? Each angle of a regular hexagon is 120°, and $3 \times 12 0^{\circ} = 36 0^{\circ}$ , so three fit round a point with no gaps. Only triangles, squares and hexagons tile a flat surface on their own, and hexagons need the least wall for the space they hold.

Each Angle of a Regular Polygon

The most common 2-mark question: find one angle of a regular polygon. Two formulas do it.

Formula

exterior angle = \frac{36 0 ^{\circ}}{n}

$n$ is the number of sides. All $n$ exterior angles are equal and add up to 360°, so each one is 360° shared by $n$ .

Formula

interior angle = \frac{( n - 2 ) \times 18 0 ^{\circ}}{n}

The top is the angle sum of all the corners. Dividing by $n$ shares it equally, because every corner is the same. You can also use $18 0^{\circ} - \frac{36 0 ^{\circ}}{n}$ : it gives the same answer.

Find one interior angle of a regular polygon with 30 sides.

Worked example

1
$360 \div 30 = 1 2^{\circ}$ — The exterior angle: 30 equal turns make 360°.
2
$180 - 12 = 16 8^{\circ}$ — Interior and exterior sit on a straight line, so they add up to 180°.
3
$(30 - 2) \times 180 \div 30 = 5040 \div 30 = 16 8^{\circ}$ — Check with the other formula: the angle sum shared between 30 equal corners. Same answer.

Key idea

Write the first line, such as $180 - 360 \div 30$ . That line earns a method mark on its own. In a “show that” question the answer is given, so the working is all the marks.

Common mistake

Rounding too early. A regular 11-sided polygon has exterior angle 360 ÷ 11 = 32.7272…°. Keep every digit: 180 − 32.7272… = 147.2727…, which is 147.3° to 1 decimal place.

How Many Sides?

The 2 or 3-mark question: each interior angle of a regular polygon is $15 6^{\circ}$ . How many sides? Go through the exterior angle.

Worked example

1
$180 - 156 = 2 4^{\circ}$ — At each corner the interior and the exterior angle sit on a straight line, so they add up to 180°.
2
$n = 360 \div 24$ — All the exterior angles add up to 360°, and each one is 24°. So count how many 24s fit into 360.
3
$n = 15$ — 360 ÷ 24 = 15, so the polygon has 15 sides. Check: 15 × 24 = 360.

Common mistake

Dividing 360 by the interior angle. $360 \div 156$ is not even a whole number.

Key idea

If you are given the exterior angle, skip the first line: $2 0^{\circ}$ gives $360 \div 20 = 18$ sides.

Interior and Exterior With x

Some questions write the interior and exterior angle with $x$ . Use interior + exterior = 180° to find $x$ , then the exterior angle gives the sides.

At each corner of a regular polygon, the interior angle is $(9 x + 16)^{\circ}$ and the exterior angle is $(x + 4)^{\circ}$ . How many sides does it have?

Worked example

1
$(9 x + 16) + (x + 4) = 180$ — The interior and exterior angle sit on a straight line, so they add up to 180°. Brackets keep each angle together.
2
$10 x + 20 = 180$ — Collect the x terms, 9x + x = 10x, then the numbers, 16 + 4 = 20.
3
$10 x = 160$ — Take 20 from both sides, so the x term is alone.
4
$x = 16$ — Divide both sides by 10.
5
$exterior = 16 + 4 = 2 0^{\circ}$ — Put x = 16 into the exterior angle. That is the one that leads to the sides.
6
$n = 360 \div 20 = 18$ — The exterior angles add up to 360°, so the polygon has 18 sides.
7
$9 \times 16 + 16 = 160, 160 + 20 = 180$ — Check: the interior angle is 160°, and 160 + 20 = 180.

Key idea

A ratio works the same way. If interior : exterior = 7 : 1, the 180° is cut into 7 + 1 = 8 parts. The exterior angle is 180 ÷ 8 = 22.5°, so $n = 360 \div 22.5 = 16$ .

Common mistake

Setting interior + exterior = 360. One corner's two angles make a straight line, 180°. It is all the exterior angles together that make 360°.

A Missing Angle in Any Polygon

The 3 or 4-mark question: the polygon is not regular. Five angles of a hexagon are $12 4^{\circ}, 10 1^{\circ}, 11 6^{\circ}, 12 1^{\circ}$ and $14 6^{\circ}$ . Find the sixth, $x$ .

Worked example

1
$(6 - 2) \times 180 = 72 0^{\circ}$ — A hexagon has 6 sides, so it splits into 4 triangles. That makes the angle sum 4 × 180°.
2
$124 + 101 + 116 + 121 + 146 = 608$ — Add the five angles you know.
3
$x = 720 - 608$ — All six angles must make 720°, so x is the part that is left.
4
$x = 11 2^{\circ}$ — Check: 608 + 112 = 720.

Common mistake

Using $36 0^{\circ}$ for every shape. Only a quadrilateral adds to $36 0^{\circ}$ . Count the sides first.

Key idea

A corner can be reflex (bigger than $18 0^{\circ}$ ). It still counts in the sum.

Key idea

Exterior angles add up to 360° in any polygon, regular or not. Four exterior angles of a pentagon are 85°, 60°, 70° and 90°. They add up to 305°, so the fifth is 360 − 305 = 55°.

Angle Sum With x or a Ratio

When the angles of a polygon are written with $x$ or given as a ratio, add them and set the total equal to the angle sum.

The angles of a pentagon are $(x + 10)^{\circ}, 2 x^{\circ}, (2 x + 15)^{\circ}, (3 x - 25)^{\circ}$ and $2 x^{\circ}$ . Find the largest angle.

Worked example

1
$(5 - 2) \times 180 = 54 0^{\circ}$ — A pentagon splits into 3 triangles, so its angles add up to 540°.
2
$(x + 10) + 2 x + (2 x + 15) + (3 x - 25) + 2 x = 540$ — Add all five angles and set the total equal to the angle sum.
3
$10 x = 540$ — Collect the x terms: 1 + 2 + 2 + 3 + 2 = 10. The numbers cancel: +10 + 15 − 25 = 0. Keep the minus on the 25.
4
$x = 54$ — Divide both sides by 10.
5
$3 \times 54 - 25 = 13 7^{\circ}$ — The largest angle is the one with the most x's, 3x − 25. Put x = 54 in.
6
$64 + 108 + 123 + 137 + 108 = 540$ — Check: the five angles add up to 540°.

Key idea

Angles in a ratio: share the angle sum. A hexagon's angles are in the ratio 3 : 4 : 4 : 5 : 5 : 3. That is 24 parts sharing 720°, so one part is 30° and the largest angle is 5 × 30 = 150°.

Key idea

Given the angle sum, find the sides: solve $(n - 2) \times 180 = 1620$ . Then $n - 2 = 9$ , so $n = 11$ . If the division does not give a whole number, such as 1000 ÷ 180 = 5.55…, no polygon has that sum.

Two Shapes Meet at a Corner

The 4 or 5-mark question: two regular polygons share a side. A regular octagon and a regular hexagon share the side $A B$ . $C$ is the next corner of the octagon after $B$ , and $D$ is the next corner of the hexagon after $B$ . Find angle $B D C$ .

Worked example

1
$360 \div 8 = 4 5^{\circ}$ — Exterior angle of the octagon: its 8 equal exterior angles add up to 360°.
2
$180 - 45 = 13 5^{\circ}$ — Interior angle of the octagon: interior + exterior = 180°.
3
$360 \div 6 = 6 0^{\circ}$ — Exterior angle of the hexagon: 6 equal exterior angles make 360°.
4
$180 - 60 = 12 0^{\circ}$ — Interior angle of the hexagon, the same way.
5
$∠ C B D = 360 - 135 - 120$ — Three angles meet at B and fill a full turn. Angles at a point add up to 360°.
6
$∠ C B D = 10 5^{\circ}$ — 360 − 135 − 120 = 105.
7
$B C = B D$ — Both are sides of polygons that share AB, so they are the same length. Triangle BCD is isosceles.
8
$∠ B D C + ∠ B C D = 180 - 105 = 7 5^{\circ}$ — Angles in a triangle add up to 180°. The two base angles share what is left.
9
$∠ B D C = 75 \div 2 = 37. 5^{\circ}$ — Base angles of an isosceles triangle are equal, so halve 75.

Common mistake

Missing the isosceles triangle. Equal sides of the two polygons make two equal angles.

Key idea

Write the reason next to each line, in words. Exam marks need them.

An Angle Inside a Regular Polygon

The 3-mark question: a line joins two corners. In a regular pentagon $A B C D E$ , find angle $A C D$ .

Worked example

1
$360 \div 5 = 7 2^{\circ}$ — Exterior angle: the 5 equal exterior angles add up to 360°.
2
$∠ A B C = 180 - 72 = 10 8^{\circ}$ — Interior angle: interior + exterior = 180°.
3
$A B = B C$ — Sides of a regular polygon are equal, so triangle ABC is isosceles.
4
$∠ B A C + ∠ B C A = 180 - 108 = 7 2^{\circ}$ — Angles in a triangle add up to 180°.
5
$∠ B C A = 72 \div 2 = 3 6^{\circ}$ — The base angles are equal, so they share the 72° equally.
6
$∠ A C D = 108 - 36 = 7 2^{\circ}$ — The whole corner at C is 108°. The line AC cuts off 36°, and ACD is the rest.

Common mistake

Thinking the line cuts the corner in half. It does not: 36° and 72° are different.

From the centre. Lines from the centre $O$ to every corner cut a regular polygon into equal isosceles triangles. Find angle $O A B$ in a regular decagon (10 sides).

Worked example

1
$∠ A O B = 360 \div 10 = 3 6^{\circ}$ — The 10 equal angles at the centre make a full turn, so each is 360° ÷ 10.
2
$O A = O B$ — The centre is the same distance from every corner, so triangle OAB is isosceles.
3
$∠ O A B + ∠ O B A = 180 - 36 = 14 4^{\circ}$ — Angles in a triangle add up to 180°.
4
$∠ O A B = 144 \div 2 = 7 2^{\circ}$ — Base angles are equal. Check: 2 × 72 = 144, the interior angle of a regular decagon.

Which Method? Read the Question

Is the polygon regular? Are you given an angle or the number of sides?

You see	You do
Regular, find the number of sides	exterior angle, then 360 ÷ it
Regular, find each angle	360 ÷ n, then 180 − it
From the centre O	360 ÷ n at the centre, then an isosceles triangle
Not regular, one angle missing	sum $(n - 2) \times 180$ , take away
Two shapes share a side	interior angles, then 360 at the point
Angles written with x	add them, set equal to the sum, solve
Interior and exterior written with x	add them to 180, solve, then 360 ÷ exterior
Angles in a ratio	add the ratio numbers, share the angle sum
Angle sum given, find the sides	solve (n − 2) × 180 = the sum
Only four sides	Angles in a Quadrilateral
Naming the shape	Polygons

Key idea

Each exterior angle of a regular polygon must divide 360 exactly. If it does not, no such polygon exists.

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