IGCSE Maths notes, topic by topic

By Fatema Bhatt, Founder, STEM-Tutor · Updated October 2026

These notes summarise each area of Cambridge IGCSE Mathematics (0580): the key methods, the facts to remember and the mistakes to avoid. They cover Extended content as well as Core. Use them with past papers, not instead of them.

1. Number

• Types of number: integers, primes, factors, multiples, squares, cubes, rational and irrational numbers. Prime factorisation finds the HCF and LCM.
• Fractions, decimals and percentages: convert between them; find percentage change = change ÷ original × 100; reverse percentages divide by the multiplier.
• Ratio and proportion: share in a ratio by adding the parts; direct proportion y = kx; inverse proportion y = k / x.
• Indices and standard form: a^m × aⁿ = a^m+n, a⁰ = 1, a⁻ⁿ = 1 / aⁿ, and fractional indices are roots. Standard form is A × 10ⁿ with 1 ≤ A < 10.
• Accuracy: round to decimal places or significant figures; upper and lower bounds are half a unit either side.
• Money and time: simple and compound interest, exchange rates, and the 24-hour clock.
• Surds (Extended): simplify √12 = 2√3 and rationalise denominators.

2. Algebra and graphs

• Expressions: expand brackets, factorise (common factors, quadratics, difference of two squares) and simplify algebraic fractions.
• Equations: linear equations, simultaneous equations (elimination or substitution), and quadratics by factorising, completing the square or the quadratic formula.
• Inequalities: solve like equations, but reverse the sign when multiplying or dividing by a negative; show regions on graphs.
• Sequences: the nth term of linear sequences is an + b; recognise quadratic, cubic and geometric sequences.
• Functions (Extended): function notation, composite functions fg(x) and inverse functions f⁻¹(x).
• Graphs: straight lines, quadratics, cubics, reciprocals and exponentials; solve equations graphically; find gradients of curves with tangents.
• Differentiation (Extended): differentiate axⁿ, find gradients and turning points.

3. Coordinate geometry

• Straight lines: y = mx + c, where m is the gradient and c the y-intercept.
• Gradient = change in y ÷ change in x; parallel lines have equal gradients; perpendicular gradients multiply to −1 (Extended).
• Length and midpoint of a line segment between two points.

4. Geometry

• Angles: angles on a straight line sum to 180°, around a point to 360°; alternate and corresponding angles with parallel lines; interior angles of a polygon sum to (n − 2) × 180°.
• Circle theorems: the angle in a semicircle is 90°; a tangent meets a radius at 90°; Extended adds angles at the centre, in the same segment and in a cyclic quadrilateral.
• Similarity and congruence: similar shapes have lengths in ratio k, areas k² and volumes k³.
• Constructions and symmetry: using a ruler and compasses, line and rotational symmetry.

5. Mensuration

• Units: convert lengths, areas (1 m² = 10 000 cm²) and volumes (1 m³ = 1 000 000 cm³).
• Areas and perimeters of rectangles, triangles, parallelograms, trapeziums and circles; arcs and sectors.
• Surface area and volume of prisms, cylinders, pyramids, cones and spheres. Many formulas are on page 2 of the paper; see the IGCSE Maths formula sheet.

6. Trigonometry

• Pythagoras' theorem and SOHCAHTOA in right-angled triangles; bearings measured clockwise from north.
• Extended: the sine rule, the cosine rule, area = ½ab sin C, graphs of sin, cos and tan, and 3D problems.

7. Transformations and vectors

• Transformations: reflections, rotations, translations and enlargements, described fully (for example, the centre, angle and direction of a rotation).
• Vectors: column vectors, adding and multiplying by a scalar, and (Extended) magnitudes and vector geometry.

8. Probability and 9. Statistics

• Probability: P(not A) = 1 − P(A); tree diagrams and Venn diagrams; independent events multiply; Extended includes conditional probability.
• Statistics: mean, median, mode and range; averages from frequency tables; bar charts, pie charts, scatter diagrams and correlation; Extended adds cumulative frequency, quartiles and histograms with unequal widths.

A reverse percentage

After a 15% increase, a price is 345 dollars. What was the original price?

• The multiplier is 1.15, so original = 345 ÷ 1.15.
• Original = 300 dollars.

A quadratic by factorising

Solve x² − 5x − 14 = 0.

• Find two numbers that multiply to −14 and add to −5: −7 and 2.
• (x − 7)(x + 2) = 0, so x = 7 or x = −2.

Similar shapes

Two similar cones have heights 6 cm and 9 cm. The smaller has volume 80 cm³. Find the larger volume.

• Length scale factor k = 9/6 = 1.5, so volume factor = 1.5³ = 3.375.
• Larger volume = 80 × 3.375 = 270 cm³.

How to use these notes

1. Read a section, then try questions on it from the 0580 past papers.
2. Practise each topic without a calculator as well as with one: Papers 1 and 2 are non-calculator.
3. Keep a list of methods you forget, and test yourself on it weekly.

Common mistakes

• Finding a percentage of the new value in reverse percentage questions.
• Forgetting to reverse an inequality sign when dividing by a negative.
• Using the length scale factor for areas or volumes in similar shapes.
• Describing a transformation without all its details.

Practice questions

Try each one before opening the answer.

Frequently asked questions

Do these notes cover Extended?

Yes. Extended-only content is marked, so Core candidates can skip it.

Is there a formula sheet in IGCSE Maths?

Yes, from 2025 a list of formulas is printed on page 2 of every 0580 paper, but not every formula is on it.

What is the best way to revise?

Short topic reviews with these notes, then timed past papers marked with the official mark schemes.

Which IGCSE Maths syllabus do these notes cover?

Cambridge IGCSE Mathematics (0580), Core and Extended. The Extended-only content is marked, so Core candidates can skip it.

Key takeaways

• Nine areas from number to statistics.
• Extended adds functions, surds, differentiation and more.
• Half the marks are non-calculator.
• Learn what is not on the formula list.
• Practise every topic with past papers.

Related guides

• IGCSE Maths formula sheet
• IGCSE Physics notes
• The quadratic formula
• Circle theorems