What is a polygon in the context of Cambridge IGCSE Mathematics?

In Cambridge IGCSE Mathematics, a polygon is defined as a two-dimensional geometric figure with straight sides. Polygons are closed shapes with a minimum of three sides, such as triangles, quadrilaterals, and pentagons. Understanding polygons is crucial for solving various geometry problems related to angles, area, and perimeter.

How do you classify polygons based on their sides and angles?

Polygons are classified based on the number of their sides and angles. Regular polygons have all sides and angles equal, such as equilateral triangles and squares. Irregular polygons have sides and angles that are not equal. Additionally, polygons can be classified by the number of sides: triangles (3 sides), quadrilaterals (4 sides), pentagons (5 sides), hexagons (6 sides), and so on.

What is the formula for calculating the sum of interior angles of a polygon?

The sum of the interior angles of a polygon can be calculated using the formula: (n - 2) × 180°, where 'n' represents the number of sides in the polygon. This formula is essential for solving problems related to angle measures in polygons, which is a key part of the Cambridge IGCSE Mathematics curriculum.

How can you determine if a polygon is convex or concave?

A polygon is convex if all its interior angles are less than 180°, meaning no line segment between any two points on the polygon's boundary passes outside the polygon. Conversely, a polygon is concave if at least one interior angle is greater than 180°, indicating that part of the polygon 'caves in'. Recognizing these properties helps in understanding the shape and structure of polygons in geometry.

What are some real-life applications of understanding polygons in geometry?

Understanding polygons is crucial in various real-life applications, including architecture, engineering, and computer graphics. For instance, architects use polygons to design floor plans and structures, while computer graphics rely on polygons to create digital images and animations. Mastery of polygons in Cambridge IGCSE Mathematics provides a foundation for these practical applications.

Why is "Using $(n - 2) \times 180$ for ONE interior angle (regular)" a common mistake in Polygons?

Why it happens: Forgetting to divide by n for an individual angle. How to avoid it: (n - 2) × 180 is the SUM. Divide by n for each angle in a regular polygon. Source: 0580 examiner reports — recurring

Why is "Applying the regular-polygon formula to an irregular one" a common mistake in Polygons?

Why it happens: Students assume the formula always works. How to avoid it: (n-2) × 180/n ONLY works when all interior angles are equal.

Why is "Multiplying the exterior-angle sum by $n$" a common mistake in Polygons?

Why it happens: Confusing per-angle with sum. How to avoid it: Sum of exterior angles is **always** 360 — independent of n.

Why is "Reporting a non-integer answer for the number of sides" a common mistake in Polygons?

Why it happens: Arithmetic slip earlier. How to avoid it: n must be a positive integer (≥ 3). If you get 5.5 or 7.2, recheck the arithmetic.

GeometryCambridge IGCSE Mathematics (0580)

Polygons

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Polygons — Cambridge IGCSE 0580 Maths Extended (2026)

Triangles, quadrilaterals, pentagons and beyond. Sum of interior angles, regular-polygon angles, and the named quadrilateral properties Cambridge expects you to recall.

What you’ll learn

Mapped to the Cambridge IGCSE 0580 syllabus (2025-2027).

E4.2 — Calculate unknown angles using interior and exterior angles of polygons.
Recognise and use the properties of common quadrilaterals.

Sum of interior angles

$(n - 2) \times 180\degree$ . Derive by triangulating from one vertex.

Formula. For an $n$ -sided polygon (an $n$ -gon), the sum of the interior angles is $(n - 2) \times 180\degree.$

Why? Pick any vertex and draw diagonals to all other vertices. The polygon splits into $(n - 2)$ triangles, each contributing $180\degree$ .

Worked. Sum of interior angles of an octagon ( $n = 8$ ):

$(8 - 2) \times 180 = 1080\degree$ .

Worked. A polygon has interior angles summing to $1440\degree$ . How many sides?

$(n - 2) \times 180 = 1440 \Rightarrow n - 2 = 8 \Rightarrow n = 10$ . Decagon.

Memorise these.

Polygon	Sides	Sum of interior angles
Triangle	3	$180\degree$
Quadrilateral	4	$360\degree$
Pentagon	5	$540\degree$
Hexagon	6	$720\degree$
Heptagon	7	$900\degree$
Octagon	8	$1080\degree$
Decagon	10	$1440\degree$

$(n - 2) \times 180\degree$ .
Memorise sums up to octagon.
Reverse: solve for $n$ if given the sum.
Formula derives from triangulation.

Regular polygons

All sides AND angles equal. Each interior angle = sum / $n$ .

A regular polygon has all sides equal AND all angles equal.

Each interior angle of a regular $n$ -gon: $\dfrac{(n - 2) \times 180\degree}{n}.$

Each exterior angle of any regular $n$ -gon: $\dfrac{360\degree}{n}.$

Worked. Each interior angle of a regular hexagon:

$\dfrac{(6 - 2) \times 180}{6} = \dfrac{720}{6} = 120\degree$ .
Each exterior angle: $\dfrac{360}{6} = 60\degree$ .
Check: interior + exterior $= 180\degree$ at each vertex ✓.

Worked. A regular polygon has each interior angle $= 156\degree$ . How many sides?

Each exterior $= 180 - 156 = 24\degree$ .
$n = \dfrac{360}{24} = 15$ . So 15 sides.

Useful relation. At every vertex: interior + exterior $= 180\degree$ (linear pair).

Regular: all sides AND all angles equal.
Each interior $= \dfrac{(n - 2) \times 180}{n}$ .
Each exterior $= \dfrac{360}{n}$ .
Interior + exterior $= 180\degree$ .

Named quadrilaterals

Memorise the properties: sides, angles, diagonals, symmetry. Cambridge tests on sight.

Square.

4 equal sides, 4 right angles.
Both pairs of sides parallel.
Diagonals equal, bisect at $90\degree$ , both diagonals are axes of symmetry.

Rectangle.

Opposite sides equal and parallel.
4 right angles.
Diagonals EQUAL and bisect each other (but not at $90\degree$ ).

Rhombus.

4 equal sides; opposite sides parallel.
Opposite angles equal.
Diagonals bisect each other at $90\degree$ (perpendicular bisectors).

Parallelogram.

Both pairs of opposite sides parallel and equal.
Opposite angles equal; co-interior angles sum to $180\degree$ .
Diagonals bisect each other (NOT at $90\degree$ , NOT equal).

Trapezium.

ONE pair of parallel sides.
An isosceles trapezium has the other two sides equal — diagonals are equal and the base angles match.

Kite.

Two pairs of ADJACENT (consecutive) sides equal.
One pair of opposite angles equal (the non-vertex pair).
Diagonals are PERPENDICULAR; one bisects the other.

Cambridge tip. When asked to identify a quadrilateral, list the properties you observe and match against the table above.

Square ⊂ rectangle ⊂ parallelogram.
Rhombus ⊂ parallelogram.
Diagonals: equal in rectangle/square; perpendicular in rhombus/kite/square.
Trapezium: only ONE pair of parallel sides.

How it’s examined

Polygon questions appear on most papers. Paper 2: 1-2 mark interior-angle calculations. Paper 4: multi-step problems combining polygon angle sums with other geometry (4-6 marks). Examiner reports flag forgetting that exterior angle sum is always $360\degree$ regardless of $n$ .

Sources: Cambridge IGCSE Mathematics 0580 syllabus 2025-2027 (E4.2); 0580/22 May/Jun 2024 — Q9 (regular polygon angle); 0580 Examiner Reports 2022-2024. Last reviewed 2026-05-05.

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Worked examples, formulae, definitions and the mistakes examiners flag — everything you need to push from a pass to an A*.

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Step-by-step worked examples — Polygons

Step-by-step solutions to past-paper-style questions on polygons, written exactly the way a tutor would explain them at the board.

1Sum of interior angles
Core• interior angles
Question
Find the sum of the interior angles of an octagon.
Step-by-step solution
1. Step 1
  $(n - 2) \times 180\degree$ with $n = 8$ .
  $(8 - 2) \times 180 = 1080$
Answer
$1080\degree$
2Each interior angle of a regular polygon
Extended• Adapted from 0580/22 May/Jun 2024 Q6• regular polygon
Question
Find the size of each interior angle of a regular decagon.
Step-by-step solution
1. Step 1
  $n = 10$ . Sum of interior angles.
  $(10 - 2) \times 180 = 1440\degree$
2. Step 2
  Each interior angle.
  $\dfrac{1440}{10} = 144\degree$
Answer
$144\degree$
3Find $n$ from each exterior angle
Extended• exterior angle
Question
Each exterior angle of a regular polygon is $24\degree$ . How many sides does the polygon have?
Step-by-step solution
1. Step 1
  Sum of exterior angles is always $360\degree$ .
  $n = \dfrac{360}{24} = 15$
Answer
$15$ sides
Examiner tip
Exterior angles of any convex polygon ALWAYS sum to $360\degree$ , regardless of the number of sides — this gives the quickest route from exterior angle to $n$ .
4Mixed polygon angle question
Extended• polygon
Question
A regular polygon has interior angles of $150\degree$ . Find the number of sides.
Step-by-step solution
1. Step 1
  Exterior angle = $180 - 150 = 30\degree$ .
2. Step 2
  $n = \dfrac{360}{30}$ .
  $n = 12$
Answer
$12$ sides

Key Formulae — Polygons

The formulae you need to memorise for polygons on the Cambridge IGCSE 0580 paper, with every variable defined in plain English and a note on when to use it.

Sum of interior angles
$(n - 2) \times 180\degree$
$n$
number of sides
When to use
For ANY $n$ -sided convex polygon.
Each interior angle (regular)
$\dfrac{(n - 2) \times 180}{n}$
When to use
Only for REGULAR polygons (all sides and angles equal).
Sum of exterior angles
$= 360\degree$
When to use
For any convex polygon — regular or irregular.
Each exterior angle (regular)
$\dfrac{360}{n}$
When to use
Only for regular polygons.

Key Definitions and Keywords — Polygons

Definitions to memorise and the exact keywords mark schemes credit for polygons answers — sharpened from recent examiner reports for the 2026 0580 sitting.

Polygon
A closed plane figure made up of straight-line segments.
Regular polygon
Examiner keyword
A polygon with all sides equal AND all interior angles equal.
Interior angle
Examiner keyword
The angle inside the polygon at each vertex.
Exterior angle
Examiner keyword
The angle between an extended side and the next side. Interior + exterior = $180\degree$ at each vertex.
Convex polygon
Every interior angle is less than $180\degree$ . All Cambridge IGCSE polygon questions assume convex unless stated otherwise.

Common Mistakes and Misconceptions — Polygons

The traps other students keep falling into on polygons questions — taken from recent Cambridge IGCSE 0580 examiner reports and mark schemes — and how to avoid them.

✕Using $(n - 2) \times 180$ for ONE interior angle (regular)
0580 examiner reports — recurring
Why it happens
Forgetting to divide by $n$ for an individual angle.
How to avoid it
$(n - 2) \times 180$ is the SUM. Divide by $n$ for each angle in a regular polygon.
✕Applying the regular-polygon formula to an irregular one
Why it happens
Students assume the formula always works.
How to avoid it
$\dfrac{(n-2) \times 180}{n}$ ONLY works when all interior angles are equal.
✕Multiplying the exterior-angle sum by $n$
Why it happens
Confusing per-angle with sum.
How to avoid it
Sum of exterior angles is always $360\degree$ — independent of $n$ .
✕Reporting a non-integer answer for the number of sides
Why it happens
Arithmetic slip earlier.
How to avoid it
$n$ must be a positive integer ( $\geq 3$ ). If you get $5.5$ or $7.2$ , recheck the arithmetic.

Practice questions

Exam-style questions with step-by-step worked solutions. Try one before checking the method.

Past paper style quiz

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Video lesson

Short walkthrough of the concepts students most often get stuck on.

Polygons — frequently asked questions

The things students keep getting wrong in this sub-topic, answered.

Polygons

Short Study Notes in the form of Slides

Detailed Study Notes

What you’ll learn

How it’s examined

1Sum of interior angles

2Each interior angle of a regular polygon

3Find nnn from each exterior angle

4Mixed polygon angle question

Sum of interior angles

Each interior angle (regular)

Sum of exterior angles

Each exterior angle (regular)

Polygon

Regular polygon

Interior angle

Exterior angle

Convex polygon

✕Using (n−2)×180(n - 2) \times 180(n−2)×180 for ONE interior angle (regular)

✕Applying the regular-polygon formula to an irregular one

✕Multiplying the exterior-angle sum by nnn

✕Reporting a non-integer answer for the number of sides

Practice questions

Past paper style quiz

Assess your understanding

Video lesson

Polygons — frequently asked questions

3Find $n$ from each exterior angle

✕Using $(n - 2) \times 180$ for ONE interior angle (regular)

✕Multiplying the exterior-angle sum by $n$