IGCSE Maths 0580 Formula Sheet with Worked Examples
Written by Dr Asbat Habib, PhD in Mathematics, 5 years of online teaching experience · Teacher Online Tutoring Academy · Updated 19 October 2026
A list of formulas helps only if you know when to use each one. This guide gives you every formula tested in Cambridge IGCSE Mathematics 0580, shows a fully worked example for the important ones, lists the mistakes we see students make most, and ends with practice questions and answers. It covers both Core and Extended, and marks which is which.
1. Area, surface area and volume
| Shape | Formula | Level |
|---|---|---|
| Triangle | A = ½bh | Core |
| Circle area / circumference | A = πr², C = 2πr | Core |
| Arc length | (θ/360) × 2πr | Extended |
| Sector area | (θ/360) × πr² | Extended |
| Prism volume | V = Al (cross-section area × length) | Core |
| Cylinder | V = πr²h, curved surface = 2πrh | Core |
| Pyramid volume | V = ⅓Ah | Extended |
| Cone | V = ⅓πr²h, curved surface = πrl | Extended |
| Sphere | V = 4⁄3πr³, surface = 4πr² | Extended |
Remember that l in the cone formula is the slant height, not the vertical height. Mixing them up is the most common error in cone questions.
Worked example: cone
A cone has radius 6 cm and vertical height 8 cm. Find its volume and curved surface area.
Volume: V = ⅓ × π × 6² × 8 = 96π ≈ 301.6 cm³.
Slant height: l = √(6² + 8²) = √100 = 10 cm.
Curved surface area: π × 6 × 10 = 60π ≈ 188.5 cm².
Worked example: arc and sector
A sector has radius 10 cm and angle 72°. Arc length = (72/360) × 2π × 10 = 4π ≈ 12.57 cm. Sector area = (72/360) × π × 10² = 20π ≈ 62.83 cm². Notice that 72/360 simplifies to ⅕, which keeps the arithmetic short.
2. Algebra
Quadratic formula (Extended): for ax² + bx + c = 0, x = (−b ± √(b² − 4ac)) ÷ 2a. The part under the root, b² − 4ac, is the discriminant. If it is positive there are two solutions, if zero there is one repeated solution, and if negative there are no real solutions.
Nth term of an arithmetic sequence: a + (n − 1)d, where a is the first term and d is the common difference.
Gradient, midpoint and distance: m = (y₂ − y₁)/(x₂ − x₁); midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2); distance = √((x₂ − x₁)² + (y₂ − y₁)²).
Worked example: quadratic formula
Solve 2x² − 5x − 3 = 0. Here a = 2, b = −5, c = −3.
Discriminant = (−5)² − 4(2)(−3) = 25 + 24 = 49. So x = (5 ± 7) ÷ 4, giving x = 3 or x = −0.5.
Always put negative values in brackets when you substitute. Writing −5² instead of (−5)² gives −25 and ruins the whole answer.
3. Trigonometry and Pythagoras
| Rule | Formula | Use it when |
|---|---|---|
| Pythagoras | a² + b² = c² | Right-angled triangle, two sides known |
| SOH CAH TOA | sin = O/H, cos = A/H, tan = O/A | Right-angled triangle, one side and one angle known |
| Sine rule | a/sin A = b/sin B = c/sin C | Any triangle with a side and its opposite angle known |
| Cosine rule | a² = b² + c² − 2bc cos A | Two sides and the angle between them, or all three sides |
| Area of triangle | ½ab sin C | Two sides and the angle between them |
Worked example: cosine rule
Two sides of a triangle are 7 cm and 9 cm with a 60° angle between them. Then a² = 7² + 9² − 2(7)(9)cos 60° = 49 + 81 − 63 = 67, so a = √67 ≈ 8.19 cm.
Check your calculator is in degree mode before you start. A wrong mode is one of the easiest ways to lose a full question.
4. Percentages, interest and rates
- Percentage change = (change ÷ original) × 100.
- Compound interest: V = P(1 + r/100)ⁿ.
- Simple interest: I = Prn/100.
- Reverse percentage: divide by the multiplier. After a 20% increase, divide by 1.2.
Worked example: compound interest
$5000 is invested at 4% a year for 3 years. V = 5000 × 1.04³ = 5000 × 1.124864 = $5624.32. The interest earned is $624.32.
5. Other formulas to learn
- Speed = distance ÷ time; density = mass ÷ volume; pressure = force ÷ area.
- Mean = total ÷ number of values; probability = favourable outcomes ÷ total outcomes.
- Angle sum of a polygon = (n − 2) × 180°; each exterior angle of a regular polygon = 360° ÷ n.
- Similar shapes: if lengths scale by k, areas scale by k² and volumes by k³.
Cambridge may print a list of formulas on the paper, and the exact list can change between exam series, so always check the current 0580 syllabus on the Cambridge website. Learn everything here regardless. Knowing a formula also tells you which method a question wants.
6. Common exam mistakes
- Using height instead of slant height in cone surface area questions.
- Rounding too early. Keep full calculator values and round only the final answer to the accuracy asked, usually 3 significant figures.
- Calculator in radians during trigonometry.
- Forgetting units, or giving area in cm instead of cm².
- Not showing working. Method marks are awarded even when the final answer is wrong.
- Mixing up the sine and cosine rules. Sine rule needs a matching side-angle pair; otherwise use cosine.
- Dropping brackets around negatives in the quadratic formula.
7. Practice questions (answers hidden)
Try each question on paper first, then open the answer.
Q1. Find the volume of a sphere with radius 3 cm.
V = 4⁄3 × π × 27 = 36π ≈ 113.1 cm³.
Q2. Solve x² + 4x − 7 = 0, giving answers to 2 decimal places.
Discriminant = 16 + 28 = 44. x = (−4 ± √44) ÷ 2 = −2 ± √11, so x = 1.32 or x = −5.32.
Q3. In a triangle, angle A = 40°, angle B = 65° and side a = 8 cm. Find side b.
b = 8 × sin 65° ÷ sin 40° ≈ 11.28 cm.
Q4. After a 20% price rise, a bike costs $90. What was the original price?
90 ÷ 1.2 = $75.
Q5. Find the area of a triangle with sides 5 cm and 8 cm and an included angle of 30°.
½ × 5 × 8 × sin 30° = 10 cm².
8. A four-week plan to lock these formulas in
- Week 1: area, volume, arc and sector. Do one worked example a day.
- Week 2: algebra and the quadratic formula. Solve five equations a day.
- Week 3: trigonometry. Practise choosing between sine rule, cosine rule and SOH CAH TOA before calculating.
- Week 4: mixed past-paper questions under timed conditions, then rewrite every formula from memory.
Past papers are free on the Cambridge International website. After each paper, mark it against the mark scheme and list the formulas you missed.
Frequently asked questions
Do I need to memorise formulas for IGCSE Maths 0580?
Cambridge lists some formulas on the question paper, but many are not given, such as Pythagoras, trigonometric ratios, speed, density and percentage change. Learn them all.
What is the difference between Core and Extended formulas?
Core candidates need the basic area, volume, Pythagoras and trigonometry formulas. Extended candidates also need the quadratic formula, sine and cosine rules, sector and arc formulas, and compound interest.
How do I remember the quadratic formula?
Say it as a rhyme and practise it on ten different equations. Write a, b and c first, then substitute with brackets around negative values.
Is the compound interest formula given in the exam?
Do not assume so. Learn V = P(1 + r/100)ⁿ.
How long should I revise formulas before the exam?
Ten minutes a day for four weeks beats one long session. Write formulas from memory, check them, and redo only the ones you missed.
Our teachers run live one-to-one IGCSE Maths classes online, starting with a free assessment lesson that finds your gaps and gives you a clear plan. Read more on our homepage or email contact@teacheronlinetutoring.com.
Book a free assessmentAbout the author: Dr Asbat Habib holds a PhD in Mathematics and has taught students online for 5 years. Reviewed by the Teacher Online Tutoring Academy team. Cambridge IGCSE is a registered trademark of Cambridge University Press & Assessment, which is not affiliated with this site.