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.

Cambridge IGCSE Mathematics (0580)

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Short Notes - Polygons

Detailed Study Notes

Detailed notes on Geometry for Cambridge IGCSE Mathematics, covering key concepts, explanations, examples, and exam-focused revision points.

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.

At a glance

Sum of interior angles of an $n$ -gon: $(n - 2) \times 180\degree$ .
Regular polygon: each interior angle = $\dfrac{(n - 2) \times 180\degree}{n}$ .
Sum of EXTERIOR angles of any polygon: $360\degree$ .
Regular: each exterior angle = $\dfrac{360\degree}{n}$ .
Square: 4 equal sides, 4 right angles, diagonals equal & perpendicular.
Rhombus: 4 equal sides, diagonals bisect at $90\degree$ .
Parallelogram: opposite sides parallel + equal; opposite angles equal.
Trapezium: ONE pair of parallel sides.

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 $180\degree$ .

Diagonals from one vertex split an n-gon into (n − 2) triangles, each worth 180°.

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).

Extend a side: the exterior angle and the interior angle form a straight line, so they sum to 180°.

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.

Dashed lines show the diagonals — equal in the square and rectangle, perpendicular in the square, rhombus and kite.

Diagonals are equal in the square and rectangle; perpendicular in the square, rhombus and kite.

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

Quick recap

Sum of interior angles: $(n - 2) \times 180\degree$ .
Sum of exterior angles: $360\degree$ (any polygon).
Regular polygon: each interior = $(n-2)180/n$ , each exterior = $360/n$ .
Square / rhombus / kite: diagonals perpendicular.
Square / rectangle: diagonals equal.
Trapezium: only one pair of parallel sides.

Memorise this

Verbatim phrases and definitions Cambridge mark schemes credit.

Polygon — a closed plane shape with straight sides.
Regular polygon — all sides equal AND all angles equal.
Sum of interior angles — $(n - 2) \times 180\degree$ for an $n$ -sided polygon.
Each exterior angle of a regular polygon — $\dfrac{360\degree}{n}$ .
Quadrilateral hierarchy — square ⊂ rectangle ⊂ parallelogram; rhombus ⊂ parallelogram.

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.

Take this whole topic with you

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.

Question type:

Question patterns to master — Polygons

Almost every polygons exam question is one of these shapes. Learn to spot each one and you will always know how to start.

Direct calculation

Recognise it by

A single instruction — find the interior-angle sum, find each angle, or find the number of sides $n$ — using one polygon formula.

How to approach it

Match the formula: sum $= (n-2)\times 180\degree$ ; each interior of a regular polygon divides that by $n$ ; each exterior $= \frac{360}{n}$ . The exterior-angle route ( $180 - \text{interior}$ ) is usually the quickest path to $n$ .

Common trap

Using $(n-2)\times 180$ as a single angle instead of the sum, or applying the regular-polygon formula to an irregular polygon. Examiner reports also flag non-integer values of $n$ as a sign of an arithmetic slip.

Show that / prove

Recognise it by

The words show that or explain why — typically whether a regular polygon tessellates.

How to approach it

Find the interior angle, then test whether it divides $360\degree$ exactly: if an integer number of copies meets at a vertex it tessellates, otherwise it leaves gaps.

Common trap

Quoting the interior angle but never comparing it with $360\degree$ . Examiner reports stress the divides- $360\degree$ comparison is the heart of the argument.

Multi-step problem

Recognise it by

A polygon diagram combining the angle sum with an extra condition — parallel sides — and two unknown angles to find.

How to approach it

Use the interior-angle sum to get one equation in the two unknowns, then use the parallel-line relationship (co-interior angles) to fix one of them, and substitute back.

Common trap

Relying on the angle sum alone and ignoring the parallel-side condition. Examiner reports note both conditions together are needed to pin down the unknowns.

1Sum of interior angles
CoreDirect calculation• 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
ExtendedDirect calculation• 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
ExtendedDirect calculation• 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
ExtendedDirect calculation• 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
5Each exterior angle of a regular hexagon
CoreDirect calculation• regular polygon, exterior angle
Question
A regular hexagon has 6 sides. Find the size of each exterior angle.
Step-by-step solution
1. Step 1
  Sum of exterior angles of any convex polygon is $360\degree$ .
2. Step 2
  For a regular polygon each exterior angle is equal, so divide by $n$ .
  $\text{Each exterior} = \dfrac{360}{6} = 60\degree$
Answer
$60\degree$
Examiner tip
The examiner report flags candidates who attempt the interior-angle route then subtract from $180$ . Going via the exterior-angle sum is faster and avoids arithmetic slips.
6Find a missing angle in an irregular pentagon
ExtendedDirect calculation• Adapted from 0580/22 Oct/Nov 2023 Q9• irregular polygon, interior angles
Question
An irregular pentagon has four interior angles of $112\degree$ , $128\degree$ , $95\degree$ and $140\degree$ . Find the fifth interior angle $x$ .
Step-by-step solution
1. Step 1
  Sum of interior angles of a pentagon ( $n = 5$ ).
  $(5 - 2) \times 180 = 540\degree$
2. Step 2
  Subtract the four known angles from $540\degree$ .
  $x = 540 - (112 + 128 + 95 + 140) = 540 - 475 = 65\degree$
Answer
$x = 65\degree$
Examiner tip
The examiner report flags candidates who apply $(n-2)\times 180 / n$ here — that formula is for REGULAR polygons only. For irregular polygons use the SUM and subtract.
7Find an interior angle of a regular 12-gon
ExtendedDirect calculation• regular polygon, dodecagon
Question
Find the size of each interior angle of a regular dodecagon (12-sided polygon).
Step-by-step solution
1. Step 1
  Method via exterior angles is fastest.
  $\text{Each exterior} = \dfrac{360}{12} = 30\degree$
2. Step 2
  Interior + exterior = $180\degree$ at each vertex.
  $\text{Each interior} = 180 - 30 = 150\degree$
Answer
$150\degree$
8Find $n$ given an interior angle
ExtendedDirect calculation• Adapted from 0580/42 Feb/Mar 2024 Q8• regular polygon, find n
Question
Each interior angle of a regular polygon is $156\degree$ . Find the number of sides $n$ .
Step-by-step solution
1. Step 1
  Use the exterior angle.
  $\text{exterior} = 180 - 156 = 24\degree$
2. Step 2
  Divide $360$ by the exterior angle.
  $n = \dfrac{360}{24} = 15$
Answer
$n = 15$ sides
Examiner tip
The examiner report flags non-integer answers — if you obtain $n = 12.5$ or similar, the value $156\degree$ would not correspond to a real regular polygon, so recheck the arithmetic.
9Show that regular hexagons tessellate
ChallengeShow that / prove• tessellation, regular polygon
Question
Show that regular hexagons tessellate (tile the plane with no gaps), and explain why regular pentagons do not.
Step-by-step solution
1. Step 1
  A regular hexagon has each interior angle equal to $\dfrac{(6-2)\times 180}{6} = 120\degree$ .
2. Step 2
  At a vertex of a tessellation, the angles meeting must sum to exactly $360\degree$ .
  $3 \times 120 = 360\degree$
3. Step 3
  Three regular hexagons fit perfectly around any vertex, so they tessellate.
4. Step 4
  A regular pentagon has interior angle $108\degree$ . No integer multiple of $108$ equals $360$ ( $3 \times 108 = 324$ , $4 \times 108 = 432$ ), so regular pentagons leave gaps.
Answer
Hexagons: three meet at $3 \times 120 = 360\degree$ → tessellate. Pentagons: $108\degree$ does not divide $360$ exactly → cannot tessellate.
Examiner tip
The examiner report flags candidates who quote $108\degree$ but fail to compare it with $360$ . The key idea is that for a regular tessellation the interior angle must divide $360\degree$ exactly.
10Compound polygon with parallel-line angles
ChallengeMulti-step problem• Adapted from 0580/42 Oct/Nov 2023 Q5• polygon, parallel lines
Question
$ABCDE$ is a pentagon with $AB$ parallel to $ED$ . Given $\angle A = 124\degree$ , $\angle B = 102\degree$ and $\angle C = 138\degree$ , find $\angle D$ and $\angle E$ .
Step-by-step solution
1. Step 1
  Sum of interior angles of a pentagon: $(5-2)\times 180 = 540\degree$ .
2. Step 2
  Hence $\angle D + \angle E = 540 - 124 - 102 - 138 = 176\degree$ .
  $\angle D + \angle E = 176\degree$
3. Step 3
  Since $AB \parallel ED$ , $\angle A$ and $\angle E$ are co-interior with the transversal $AE$ , so they sum to $180\degree$ .
  $\angle E = 180 - 124 = 56\degree$
4. Step 4
  Substitute back.
  $\angle D = 176 - 56 = 120\degree$
Answer
$\angle E = 56\degree,\ \angle D = 120\degree$
Examiner tip
The examiner report flags candidates who use the angle sum alone without exploiting the parallel-line condition — both conditions together pin down the two unknowns.

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.

Polygons — frequently asked questions

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

Polygon

Sides

Sum of interior angles

Triangle

180\degree

Quadrilateral

360\degree

Pentagon

540\degree

Hexagon

720\degree

Heptagon

900\degree

Octagon

1080\degree

Decagon

1440\degree

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.

Dashed lines show the diagonals — equal in the square and rectangle, perpendicular in the square, rhombus and kite.

Diagonals are equal in the square and rectangle; perpendicular in the square, rhombus and kite.

Polygons

Short Notes - Polygons

Detailed Study Notes

At a glance

What you’ll learn

Sum of interior angles

Regular polygons

Named quadrilaterals

Quick recap

Memorise this

How it’s examined

Take this whole topic with you

Question patterns to master — Polygons

1Sum of interior angles

2Each interior angle of a regular polygon

3Find nnn from each exterior angle

4Mixed polygon angle question

5Each exterior angle of a regular hexagon

6Find a missing angle in an irregular pentagon

7Find an interior angle of a regular 12-gon

8Find nnn given an interior angle

9Show that regular hexagons tessellate

10Compound polygon with parallel-line angles

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

Polygons — frequently asked questions

Polygons

Short Notes - Polygons

Detailed Study Notes

At a glance

What you’ll learn

Sum of interior angles

Regular polygons

Named quadrilaterals

Quick recap

Memorise this

How it’s examined

Take this whole topic with you

Question patterns to master — Polygons

1Sum of interior angles

2Each interior angle of a regular polygon

3Find nnn from each exterior angle

4Mixed polygon angle question

5Each exterior angle of a regular hexagon

6Find a missing angle in an irregular pentagon

7Find an interior angle of a regular 12-gon

8Find nnn given an interior angle

9Show that regular hexagons tessellate

10Compound polygon with parallel-line angles

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

Polygons — frequently asked questions

3Find $n$ from each exterior angle

8Find $n$ given an interior angle

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

✕Multiplying the exterior-angle sum by $n$

3Find $n$ from each exterior angle

8Find $n$ given an interior angle

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

✕Multiplying the exterior-angle sum by $n$