Geometry and measures

Angles in polygons

Q: What do the angles in a hexagon add up to?

(6 - 2) \times 180 = 720^\circ. In a regular hexagon each angle is 120^\circ.

11 past-paper questions, easiest first. Try each one, then reveal the answer.

Foundation & Higher9 of 11 from Foundation papers

Revision guide by The Full Marks

Angles in polygons: what you need to know

A polygon is a flat shape with straight sides. By splitting any polygon into triangles you can find the total of its interior angles, and for regular polygons you can find each angle. Both tiers see this, often mixed with other angle facts.

Key facts

Sum of interior angles $= (n - 2) \times 180^\circ$ , where $n$ is the number of sides.
Exterior angles of any polygon add up to $360^\circ$ .
Interior angle $+$ exterior angle $= 180^\circ$ .
Regular polygon: each exterior angle $= \frac{360^\circ}{n}$ , so $n = \frac{360^\circ}{\text{exterior angle}}$ , and each interior angle $= 180^\circ -$ exterior angle.
Triangle $180^\circ$ , quadrilateral $360^\circ$ , pentagon $540^\circ$ , hexagon $720^\circ$ .

Missing angle in any polygon

Count the sides and work out $(n - 2) \times 180$ .
Add up the angles you know and subtract from the total.

Regular polygons

Find the exterior angle: $360 \div n$ .
Interior angle $= 180 -$ exterior angle.
To find $n$ , always divide 360 by the exterior angle, not the interior one.

Worked examples

Example 1 Foundation · grade 3

A pentagon has angles $100^\circ$ , $110^\circ$ , $95^\circ$ , $120^\circ$ and $x$ . Find $x$ .

Show the solution

Sum of angles $= (5 - 2) \times 180 = 540^\circ$ .
$100 + 110 + 95 + 120 = 425$
$x = 540 - 425 = 115^\circ$ .

Example 2 Higher · grade 6

Each interior angle of a regular polygon is 4 times its exterior angle. How many sides does it have?

Show the solution

Let the exterior angle be $e$ . Then the interior angle is $4e$ .
$e + 4e = 180$ , so $5e = 180$ and $e = 36^\circ$ .
$n = 360 \div 36 = 10$ .
The polygon has 10 sides (a decagon).

Common mistakes

Dividing 360 by the interior angle to find $n$ . Use the exterior angle.
Using the wrong angle sum: $180^\circ$ for a quadrilateral or $360^\circ$ for a pentagon. Work it out each time with $(n-2) \times 180$ .
Using the exterior angle (e.g. $72^\circ$ for a regular pentagon) where the diagram needs the interior angle ( $108^\circ$ ).
Using the total angle sum as if it were one angle, or forgetting a right angle marked on the diagram.

Exam tips

Write the angle sum line, e.g. $(6 - 2) \times 180 = 720$ . It shows your method clearly, and method marks are given for correct working like this.
When two regular polygons meet at a point, find each interior angle first, then use the fact that angles around a point add up to $360^\circ$ .
If a question says 'give a reason', write the fact in words, e.g. 'angles in a quadrilateral add up to $360^\circ$ '.

Questions people ask

How do you find the interior angle of a regular polygon?

Divide 360 by the number of sides to get the exterior angle, then subtract that from 180. A regular octagon has exterior $45^\circ$ and interior $135^\circ$ .

What do the angles in a hexagon add up to?

$(6 - 2) \times 180 = 720^\circ$ . In a regular hexagon each angle is $120^\circ$ .

Exam questions: Angles in polygons (11, easiest first)

Angles in polygons3 marksGrade 3CalculatorFoundation

November 2022 · Paper 3 Foundation · Q13

Find a missing angle in a quadrilateral with a reason, and explain why a triangle's labelled angles are impossible.

Show answer

(a(i))	40°	1 mark
(a(ii))	Angles in a quadrilateral add up to 360°	1 mark
(b)	$80 + 60 + 50 = 190$ , but angles in a triangle add up to 180°	1 mark

Worked solution Checked against the mark scheme

(a) $x = 360 - 120 - 120 - 80 = 40°$ because the angles in a quadrilateral add up to 360°.

(b) $80 + 60 + 50 = 190°$ . Angles in a triangle must add up to 180°, so the diagram cannot be right.

Watch out: Using 180° for the quadrilateral.

Angles in polygons3 marksGrade 3CalculatorFoundation

November 2020 · Paper 3 Foundation · Q11

Use the angle sum of a quadrilateral and angles on a straight line to find an exterior angle.

Show answer

Answer	110	3 marks

Worked solution Checked against the mark scheme

Angles in a quadrilateral add to $360°$ :

$360 - 130 - 95 - 65 = 70°$

Angles on a straight line add to $180°$ :

$y = 180 - 70 = 110$

Watch out: Stopping at 70, the interior angle.

Angles in polygons4 marksGrade 5No calculatorOn both tiers

November 2024 · Paper 1 Foundation · Q23

Use the angle sum of a pentagon with angles given in terms of one unknown to find that angle.

Show answer

Answer	$55$	4 marks

Worked solution Checked against the mark scheme

Angle sum $= (5 - 2) \times 180 = 540°$

$c + 155 + c + 3c + 2c = 540$

$7c = 385$ , so $c = 55$

Angle $a = 55°$

Watch out: Using 360° as the angle sum.

Angles in polygons3 marksGrade 5CalculatorHigher

November 2024 · Paper 2 Higher · Q9

Use the angles of two regular nine-sided polygons to prove a quadrilateral between them is not a square.

Show answer

Answer	Angle $ABC = 80°$ , not $90°$ , so ABCD is not a square	3 marks

Worked solution Checked against the mark scheme

Exterior angle $= 360 \div 9 = 40°$ , so interior angle $= 140°$ .

Angles at point B add to 360°: $360 - 140 - 140 = 80°$ .

A square has all angles 90°, but angle $ABC = 80°$ , so ABCD is not a square.

Watch out: Using the angle sum of the nonagon (1260°) as if it were one angle.

Angles in polygons4 marksGrade 5No calculatorHigher

June 2023 · Paper 1 Higher · Q10

Use the angle sum of a pentagon and a given multiple between two unknown angles to find one of them.

Show answer

Answer	$140^\circ$	4 marks

Worked solution Checked against the mark scheme

Angles in a pentagon add to $(5 - 2) \times 180 = 540^\circ$

The two unknown angles total $540 - 120 - 135 - 110 = 175^\circ$

Let angle $ABC = x$ , so angle $AED = 4x$ . Then $5x = 175$ , so $x = 35$

Angle $AED = 4 \times 35 = 140^\circ$

Watch out: Using 360° as the angle sum of a pentagon.

Angles in polygons3 marksGrade 5No calculatorOn both tiers

June 2022 · Paper 1 Foundation · Q27

Find the angle between a regular hexagon and a regular pentagon that share a side.

Show answer

Answer	132	3 marks

Worked solution Checked against the mark scheme

Hexagon interior angle: $180 - 360 \div 6 = 120^\circ$

Pentagon interior angle: $180 - 360 \div 5 = 108^\circ$

Angles round a point add to $360^\circ$ : $x = 360 - 120 - 108 = 132^\circ$

Watch out: Using exterior angles in place of interior ones and getting 228.

Angles in polygons2 marksGrade 5CalculatorFoundation

November 2020 · Paper 2 Foundation · Q28

Find the number of sides of a regular polygon from its exterior angle.

Show answer

Answer	24	2 marks

Worked solution Checked against the mark scheme

Exterior angles of any polygon add up to $360^\circ$ .

$360 \div 15 = 24$ sides.

Watch out: Dividing 180 by 15 to get 12.

Angles in polygons4 marksGrade 5CalculatorOn both tiers

November 2019 · Paper 3 Foundation · Q29

Find an angle in a diagram combining a regular pentagon and a parallelogram.

Show answer

Answer	45°	4 marks

Worked solution Checked against the mark scheme

Co-interior angles in the parallelogram add to 180°: $180 - 117 = 63°$

Interior angle of a regular pentagon: $(5-2) \times 180 \div 5 = 108°$

$x = 108 - 63 = 45°$

Watch out: Using 72° (the exterior angle) as the pentagon's interior angle.

Angles in polygons4 marksGrade 5CalculatorOn both tiers

June 2019 · Paper 3 Foundation · Q28

Use symmetry, the hexagon angle sum and a 2 : 1 angle relation to find an angle in a hexagon.

Show answer

Answer	162	4 marks

Worked solution Checked against the mark scheme

Interior angles of a hexagon: $(6 - 2) \times 180 = 720^\circ$

By symmetry $A = B = 117^\circ$ ; let $D = E = x$ and $C = F = 2x$ .

$117 + 117 + 2x + 2x + x + x = 720$ , so $6x = 486$ and $x = 81$

Angle $AFE = 2 \times 81 = 162^\circ$

Watch out: Giving 81°, the value of angle CDE, as the answer.

Angles in polygons3 marksGrade 5No calculatorFoundation

November 2018 · Paper 1 Foundation · Q28

Find the number of sides of a regular polygon whose interior angle is 11 times its exterior angle.

Show answer

Answer	24	3 marks

Worked solution Checked against the mark scheme

Interior + exterior = 180°, so $x + 11x = 180$ .

$12x = 180 \Rightarrow x = 15°$ (exterior angle)

Number of sides $= 360 \div 15 = 24$

Watch out: Dividing 360 by the interior angle.

Angles in polygons5 marksGrade 5CalculatorOn both tiers

June 2018 · Paper 3 Foundation · Q26

Use the angle sum of a pentagon with one angle twice another to find a missing angle.

Show answer

Answer	$140^\circ$	5 marks

Worked solution Checked against the mark scheme

Angle sum of a pentagon: $(5 - 2) \times 180 = 540^\circ$ .

Known angles: $125 + 115 + 90 = 330$ , so $540 - 330 = 210^\circ$ remains.

If $ABC = x$ , then $x + 2x = 210$ , so $x = 70$ .

$BCD = 2 \times 70 = 140^\circ$ .

Watch out: Forgetting the right angle at D.

Also tested in these questions

Their main topic is different, but part of each one tests this topic.

November 2023 · Paper 2 Higher · Q16: In a trapezium, use the sine rule then the sine area formula to find the area of a triangle.
June 2023 · Paper 2 Higher · Q24: Find the exact area of a star-shaped region enclosed by eight regular octagons, in surd form.
November 2022 · Paper 3 Higher · Q16: Use cyclic quadrilateral facts and an angle ratio to find an angle outside a circle.
November 2022 · Paper 3 Higher · Q26: Find a diagonal of a regular heptagon from the area of a triangle formed by two sides.
June 2022 · Paper 2 Higher · Q20: Use circle theorems and angle facts to find an angle outside a cyclic quadrilateral, giving reasons.
November 2020 · Paper 2 Higher · Q13: Find a side of a non-right-angled triangle using the sine rule after working out the missing angle.
June 2019 · Paper 2 Higher · Q18: Find an angle outside a cyclic quadrilateral using isosceles triangles and circle theorems, with reasons.
November 2018 · Paper 3 Foundation · Q25: Use an angle ratio in a right-angled triangle to find the angles, then use trigonometry to find the hypotenuse.
June 2018 · Paper 1 Higher · Q11: Use circle and triangle facts to find an angle at a tangent in terms of x, giving reasons.
Specimen · Paper 1 Foundation · Q20: Form and solve an equation from the angles in a right-angled triangle, then find the smallest angle.
Specimen · Paper 2 Higher · Q9: Use an isosceles triangle and a given bearing to find the bearing of one point from another.