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Cambridge International GCSE Mathematics (0580) — Full Specification Breakdown
Our summary of the Cambridge 0580 topics, clearly labelled as Core or Extended-only. Includes exam tips for each topic so you know exactly what Cambridge expects.
Syllabus version: Cambridge IGCSE Mathematics 0580, 2025–2027 (our summary, not the official document)
48
Core topics
36
Extended only
84
Total topics
Showing all 84 topics. Core topics (blue) appear on both Core and Extended papers. Extended Only topics (purple) appear on Extended papers only.
Identify and use natural numbers, integers (positive, negative, zero), prime numbers, square numbers, cube numbers, triangle numbers, and common factors/multiples.
Exam tip: Know your prime numbers up to 50. Questions often ask you to "write as a product of prime factors" — use a factor tree and express using index notation.
Find the highest common factor (HCF) and lowest common multiple (LCM) of two or more numbers, using prime factor decomposition where appropriate.
Exam tip: Use a Venn diagram of prime factors: HCF = intersection (multiply shared factors), LCM = union (multiply all factors). Common trap: confusing which is which.
Calculate squares, square roots, cubes, and cube roots of numbers, including non-perfect squares/cubes using a calculator.
Exam tip: Know squares up to 15² and cubes up to 5³ without a calculator. For non-perfect roots, your calculator is essential — learn the √ and ³√ functions.
Order and perform arithmetic with positive and negative numbers, including in real-world contexts (temperature, debt, altitude).
Exam tip: Remember: subtracting a negative is adding. −3 − (−5) = −3 + 5 = 2. Draw a number line when in doubt.
Convert between fractions, decimals, and percentages. Perform the four operations (+, −, ×, ÷) with fractions and mixed numbers. Order fractions by converting to a common denominator.
Exam tip: When dividing fractions, use "keep, change, flip" — keep the first fraction, change ÷ to ×, flip the second. Show all working even if you use a calculator.
Write numbers in standard form A × 10ⁿ where 1 ≤ A < 10. Convert between standard form and ordinary numbers. Perform calculations in standard form.
Exam tip: Negative powers of 10 = small numbers (e.g. 3.2 × 10⁻⁴ = 0.00032). Adding/subtracting in standard form: convert to the same power of 10 first.
Round numbers to a given number of decimal places or significant figures. Estimate calculations by rounding to 1 significant figure. Understand the effect of rounding on the accuracy of a result.
Exam tip: For estimation questions, always round every number to 1 significant figure, then calculate. Never mix rounding strategies mid-question.
Understand that measurements are subject to rounding error. Find the upper and lower bounds of rounded values and use them to find the bounds of compound measures.
Exam tip: Upper bound = rounded value + half the unit of rounding. For a maximum result: use max ÷ min or max × max as appropriate. Always state which bound you used and why.
Use ratio notation, simplify ratios, divide quantities in a given ratio. Understand direct and inverse proportion and solve problems involving them.
Exam tip: For direct proportion: if x doubles, y doubles. For inverse: if x doubles, y halves. Set up a proportion equation (y = kx or y = k/x) and find k first.
Solve problems involving rates, including speed (distance ÷ time), density (mass ÷ volume), pressure (force ÷ area), and unit conversion.
Exam tip: Speed = Distance ÷ Time. Cover up the unknown in the triangle to find the formula. Always check units — if distance is in km and time in minutes, convert to hours first.
Calculate percentages of amounts, percentage increase/decrease, percentage change, reverse percentages (finding original value), compound interest, and simple interest.
Exam tip: For reverse percentage: divide by the multiplier. E.g. price after 20% increase is £60 → original = 60 ÷ 1.2 = £50. For compound interest use Aₙ = P(1 + r/100)ⁿ.
Solve problems involving time in 12-hour and 24-hour clock formats, including timetables, elapsed time, and time zone conversions.
Exam tip: Elapsed time questions: convert both times to 24-hour format, then subtract. 60 minutes = 1 hour — avoid decimal time errors (2.5 hours ≠ 2h 50min).
Solve problems involving money, currency conversion, profit/loss, best value comparisons, and bills/invoices.
Simplify expressions involving surds (e.g. √12 = 2√3). Expand and simplify expressions with surds. Rationalise the denominator of fractions containing surds (e.g. 1/√5 = √5/5).
Exam tip: To rationalise a/√b: multiply top and bottom by √b. For a/(p + √q): multiply by (p − √q)/(p − √q) — the "conjugate". Keep answers in exact surd form unless told otherwise.
Convert recurring decimals to exact fractions using algebraic methods (e.g. 0.̄3 = 1/3, 0.1̄2̄ = 12/99).
Exam tip: Let x = 0.272727... → 100x = 27.272727... → 99x = 27 → x = 27/99 = 3/11. Multiply by 10ⁿ where n is the length of the repeating block.
Substitute numerical values into formulae and algebraic expressions, including those with multiple variables and powers.
Exam tip: Always use brackets when substituting negative numbers. e.g. for a = −3, a² = (−3)² = 9, not −9.
Expand single brackets and pairs of brackets (including perfect squares and difference of two squares). Simplify by collecting like terms.
Exam tip: For (a + b)(a − b) = a² − b². Memorise this — it saves steps on many factorisation and proof questions.
Factorise by extracting a common factor, by difference of two squares, and by grouping. Factorise quadratic trinomials of the form ax² + bx + c.
Exam tip: Always check for a common factor first before trying other methods. For trinomials with a > 1, use the AC method: find two numbers that multiply to ac and add to b.
Change the subject of a formula, including cases where the subject appears more than once or inside a root or fraction.
Exam tip: Treat it like solving an equation but in letters. If the subject appears twice: factorise it out first. E.g. ax + b = cx + d → x(a − c) = d − b → x = (d − b)/(a − c).
Solve linear equations in one unknown, including those with brackets, fractions, and unknowns on both sides.
Exam tip: When fractions are present, multiply every term by the lowest common denominator first to eliminate fractions before solving.
Translate worded problems into algebraic equations or inequalities, solve them, and interpret the solution in context.
Exam tip: Define your unknown clearly at the start: "Let x = ...". Write the equation in terms of x, solve, then re-read the question to check you answered what was asked.
Solve two simultaneous linear equations in two unknowns using elimination and substitution methods.
Exam tip: Elimination: multiply equations to match one variable, then add or subtract. Always verify your answer by substituting both values back into both original equations.
Solve linear inequalities in one variable and represent solutions on a number line. Use set notation where required.
Exam tip: When multiplying or dividing an inequality by a negative number, reverse the inequality sign. Open circle (○) = strict inequality; closed circle (●) = includes endpoint.
Generate and describe linear (arithmetic) and geometric sequences. Find the next term and the term-to-term rule. Write the nth term formula for linear sequences.
Exam tip: For linear sequences: nth term = first term + (n−1) × common difference = dn + (a−d). For geometric sequences: multiply by the common ratio each time.
Plot and interpret straight line graphs. Find the gradient and y-intercept from an equation in the form y = mx + c. Find the equation of a straight line from two points or a graph.
Exam tip: Gradient = rise ÷ run = (y₂ − y₁)/(x₂ − x₁). A positive gradient slopes upward left to right. Rearrange any equation into y = mx + c to read off m and c directly.
Interpret and draw distance–time graphs. Calculate average speed from gradient. Recognise rest (horizontal), outward, and return journeys.
Exam tip: A steeper line = faster speed. A horizontal section = stationary. Speed = gradient of the line (distance ÷ time). Units matter — read the axis labels carefully.
Understand function notation f(x). Find the domain and range of a function. Evaluate f(a) and solve f(x) = a.
Exam tip: Domain = allowed input values; range = possible output values. For f(x) = 1/(x−2), domain excludes x = 2 (division by zero). For f(x) = √x, domain is x ≥ 0.
Form and evaluate composite functions fg(x) meaning f applied after g. Understand that fg ≠ gf in general.
Exam tip: fg(x) means do g first, then f. Work from right to left: substitute g(x) into f. Always simplify fully and check by substituting a number.
Find the inverse function f⁻¹(x). Understand that f⁻¹ reverses f, so ff⁻¹(x) = x. Restrict domain if necessary for the inverse to be a function.
Exam tip: To find f⁻¹: replace f(x) with y, swap x and y, rearrange to make y the subject, then write f⁻¹(x) = ...
Solve quadratic equations using the quadratic formula x = (−b ± √(b²−4ac)) / 2a. Complete the square to write ax² + bx + c in vertex form a(x + p)² + q. Use to find max/min points and solve equations.
Exam tip: Discriminant b²−4ac: positive = two real roots, zero = one repeated root, negative = no real roots. Completing the square: always check by expanding back.
Recognise and sketch the shapes of y = ax² + bx + c (parabola), y = ax³ (cubic), y = a/x (reciprocal/hyperbola). Identify roots, turning points, and asymptotes.
Exam tip: For quadratics: roots = x-intercepts (solve = 0), vertex = turning point (x = −b/2a), y-intercept = c. Positive a = U-shape; negative a = ∩-shape.
Solve quadratic inequalities in one variable, e.g. x² − 5x + 6 > 0. Represent solutions on a number line or using set notation.
Exam tip: Sketch the parabola and identify where it is above/below the x-axis. For a positive coefficient: ax² + bx + c > 0 gives two separate intervals; < 0 gives a single interval between roots.
Find the nth term formula for quadratic sequences where the second difference is constant.
Exam tip: Second difference = 2a, where nth term = an² + bn + c. Find a first, then use known terms to build two equations in b and c. Always verify with n = 1, 2, 3.
Recognise and sketch exponential functions y = aˣ (a > 0, a ≠ 1). Understand exponential growth and decay. Solve simple exponential equations.
Exam tip: Exponential graphs always pass through (0, 1) if y = aˣ. Growth: a > 1; decay: 0 < a < 1. The x-axis is a horizontal asymptote — the graph never touches it.
Understand that log_a(x) is the inverse of aˣ. Use the definition log_a(b) = c ⟺ aᶜ = b to evaluate and solve simple logarithmic equations.
Exam tip: Read log as "the power I raise the base to in order to get...". So log₂(8) = 3 because 2³ = 8. For 0580, logarithm questions are straightforward — no log laws required beyond base definition.
Simplify algebraic fractions by factorising numerator and denominator and cancelling. Add, subtract, multiply, and divide algebraic fractions.
Exam tip: Always factorise fully before cancelling — never cancel individual terms, only common factors. For adding, find a common denominator and combine numerators.
Calculate the midpoint of a line segment given the coordinates of its endpoints using the midpoint formula: M = ((x₁+x₂)/2, (y₁+y₂)/2).
Calculate the gradient of a straight line from two points. Write and use the equation y = mx + c. Find the equation of a line given its gradient and one point, or two points.
Exam tip: Use point-slope form: y − y₁ = m(x − x₁), then rearrange to y = mx + c. Always verify by checking both points satisfy the equation.
Calculate the length of a line segment using the distance formula: d = √((x₂−x₁)² + (y₂−y₁)²), derived from Pythagoras' theorem.
Exam tip: This is Pythagoras in coordinate form. Sketch the points and draw the horizontal and vertical components to visualise before calculating.
Understand that perpendicular lines have gradients m₁ and m₂ where m₁ × m₂ = −1 (negative reciprocals). Use this to find equations of perpendicular bisectors and altitudes.
Exam tip: Perpendicular gradient = −1/m. A horizontal line (gradient 0) is perpendicular to a vertical line (undefined gradient). These pairs require different handling.
Use angle properties of parallel lines (alternate, corresponding, co-interior angles), triangles (angle sum, exterior angle), and polygons (interior and exterior angle sums: (n−2)×180° and 360°).
Exam tip: Always state the reason for each step in angle calculations: "alternate angles (parallel lines)", "angles in a triangle sum to 180°", etc. Cambridge marks reasons separately.
Identify and use line symmetry and rotational symmetry in 2D shapes. Know the order of rotational symmetry and number of lines of symmetry for common polygons.
Know and use: centre, radius, diameter, chord, tangent, arc, sector, segment. Understand that the tangent to a circle at a point is perpendicular to the radius at that point.
Exam tip: Learn all the vocabulary — questions often ask you to "name the feature" or "complete the sentence". Tangent is always at 90° to the radius — this is also a circle theorem used in extended.
Identify similar and congruent triangles. Know the four congruence conditions (SSS, SAS, AAS/ASA, RHS). Use similarity to find missing lengths and areas (length ratio k → area ratio k²).
Exam tip: Ratio of lengths = k. Ratio of areas = k². Ratio of volumes = k³. Examiners often give you areas and ask for volumes — cube the linear scale factor, don't square it.
Interpret and draw scale drawings, plans, and maps. Use map scales to find actual distances.
Construct using ruler and compasses only: the perpendicular bisector of a line segment, the angle bisector of an angle, and a 60° angle. Draw arcs accurately.
Exam tip: Always leave construction arcs visible — do not erase them. Cambridge examiners check that arcs are present to award method marks.
Construct the locus of a point satisfying a given condition: equidistant from two points (perpendicular bisector), equidistant from two lines (angle bisector), fixed distance from a point (circle), fixed distance from a line (parallel lines + semicircles at ends).
Exam tip: Shade the correct region and keep construction marks. A locus question combines construction skills — read the conditions carefully and take them one at a time.
Apply and prove: angle at the centre is twice the angle at the circumference; angle in a semicircle is 90°; angles in the same segment are equal; tangent-radius is 90°; alternate segment theorem; opposite angles in a cyclic quadrilateral sum to 180°; tangents from an external point are equal.
Exam tip: State the theorem by name for each step. "Angle in a semicircle" and "angles in same segment" are most frequently tested. Draw the radius to any tangent point immediately.
Calculate the perimeter and area of rectangles, triangles, parallelograms, trapeziums, and circles (C = 2πr, A = πr²). Combine for composite shapes.
Exam tip: Trapezium area = ½(a + b)h where a and b are the parallel sides. Always check which dimensions are perpendicular heights vs slant lengths.
Calculate the volume and total surface area of cuboids, cylinders, cones, spheres, and pyramids using standard formulae.
Exam tip: Key formulae to memorise: V(sphere) = 4/3 πr³, SA(sphere) = 4πr², V(cone) = 1/3 πr²h, SA(cone) = πrl + πr² where l = slant height. These appear on the formula sheet — always check.
Calculate the arc length and area of a sector of a circle: arc length = (θ/360) × 2πr; sector area = (θ/360) × πr².
Exam tip: θ must be in degrees for these formulae. Perimeter of a sector = arc length + 2r (two radii). Don't forget the straight edges when finding sector perimeter.
Calculate the volume of compound 3D shapes by addition or subtraction. Find the volume of a frustum (truncated cone) by subtracting the small cone from the large cone.
Exam tip: For a frustum: use similar triangles to find the height of the missing small cone, then subtract its volume. Draw a full cone and label the complete height before calculating.
Use sin, cos, and tan ratios to find missing sides and angles in right-angled triangles.
Exam tip: Label the triangle H (hypotenuse, opposite right angle), O (opposite the angle), A (adjacent, next to the angle). Cover up the unknown in the SOH-CAH-TOA triangles to get the formula.
Apply Pythagoras' theorem a² + b² = c² to find missing sides in right-angled triangles. Use in 2D and simple 3D contexts.
Exam tip: The hypotenuse is always the longest side, opposite the right angle. Subtract when finding a shorter side: a² = c² − b². Always square root at the end, not before.
Solve problems involving angles of elevation (looking up) and depression (looking down). Use and interpret three-figure bearings (000° to 360°).
Exam tip: Bearings are always measured clockwise from North. Draw a clear diagram for every bearing problem. Elevation and depression are equal for horizontal line of sight between two points.
Apply the sine rule a/sinA = b/sinB = c/sinC to find missing sides or angles in non-right-angled triangles.
Exam tip: Use sine rule when you have a matching angle-side pair. Watch for the ambiguous case: when given two sides and a non-included angle, there may be two possible triangles.
Apply the cosine rule a² = b² + c² − 2bc cosA (and its rearrangement to find angles) for non-right-angled triangles. Recognise when to use it: SAS (finding a side) or SSS (finding an angle).
Exam tip: Use cosine rule when you have SAS (two sides + included angle) or SSS (all three sides, find an angle). It's harder to use than sine rule — check: do you have a matching pair? If not, use cosine.
Calculate the area of any triangle using the formula Area = ½ab sinC where a and b are two sides and C is the included angle.
Exam tip: Use this when you know two sides and the included angle but not the height. It's often combined with the sine or cosine rule in multi-part questions.
Solve problems involving right-angled and non-right-angled triangles in 3D contexts, including finding angles between a line and a plane.
Exam tip: Identify the 2D cross-section triangle within the 3D shape, extract its measurements, then apply the appropriate trig rule. Skew and diagonal problems always reduce to a 2D triangle.
Sketch and interpret the graphs of y = sin x, y = cos x, and y = tan x for 0° ≤ x ≤ 360°. Identify amplitude, period, and key values.
Exam tip: sin and cos have period 360° and amplitude 1. tan has period 180° with asymptotes at 90° and 270°. Know the key values: sin 30° = cos 60° = 0.5, sin 90° = 1, cos 0° = 1.
Reflect shapes in horizontal lines (y = a), vertical lines (x = b), and the lines y = x and y = −x. Describe reflections fully by stating the mirror line.
Exam tip: Count the perpendicular distance from each vertex to the mirror line, then plot the same distance on the other side. The image is the same size and shape — congruent.
Rotate shapes by 90°, 180°, and 270° about a given centre (not necessarily the origin). Describe rotations fully: angle, direction, centre of rotation.
Exam tip: For 90° clockwise: (x, y) → (y, −x). For 90° anticlockwise: (x, y) → (−y, x). For 180°: (x, y) → (−x, −y). Use tracing paper in the exam if allowed.
Translate shapes using a column vector [[a],[b]]. Describe translations by stating the column vector.
Exam tip: Add the column vector to every vertex: new x = old x + a, new y = old y + b. Positive a moves right; positive b moves up.
Enlarge a shape with a positive integer scale factor from a given centre. Describe enlargements by stating the scale factor and centre.
Exam tip: Distances from the centre of enlargement scale by the factor. Draw lines through the centre and through each vertex, then measure along those lines. All lengths multiply by k.
Enlarge shapes using fractional scale factors (produces a smaller image) and negative scale factors (produces an image on the opposite side of the centre, inverted).
Exam tip: Negative scale factor: measure from centre to vertex, then go the same distance in the opposite direction (through the centre). The image is inverted and on the other side.
Apply two successive transformations and describe the single equivalent transformation. Understand that the order of transformations matters (they are not generally commutative).
Exam tip: Work step by step: apply first transformation to get image A'B'C', then apply second to A'B'C' to get A''B''C''. Then describe the single transformation that maps the original to the final image.
Identify which transformation maps one shape to another and state all required information fully (e.g. for rotation: angle, direction, centre; for enlargement: scale factor, centre).
Exam tip: Scoring rubric: each piece of information (type, details) is marked separately. Never write two types of transformation — pick exactly one. "Rotation and translation" scores zero.
Write vectors using column notation [[a],[b]] and bold/arrow notation. Calculate the magnitude (length) of a vector using |v| = √(a² + b²).
Exam tip: Magnitude is always positive. |[[3],[-4]]| = √(9+16) = √25 = 5. This is Pythagoras in vector form.
Add and subtract vectors. Multiply a vector by a scalar. Understand parallel vectors (one is a scalar multiple of the other).
Exam tip: To go from A to C via B: AC = AB + BC. Reversing a vector negates it: BA = −AB. Scalar multiplication stretches or shrinks the vector.
Understand position vectors OA, OB etc. measured from the origin O. Express other vectors in terms of position vectors: AB = OB − OA.
Exam tip: Think of position vectors as coordinates in vector form. AB = b − a (position vector of B minus position vector of A). Draw a diagram to confirm direction.
Use vectors to prove geometric results: collinearity of three points, midpoints, parallel lines, and properties of shapes.
Exam tip: Collinearity: show that AC = k × AB (same direction) for some scalar k. State this explicitly. For midpoints: use the midpoint formula with position vectors: M = ½(a + b).
Use the probability scale 0 to 1. Calculate probabilities from sample spaces. Understand theoretical vs experimental probability. Use P(A) + P(A') = 1.
Exam tip: Probability = favourable outcomes ÷ total possible outcomes. Always simplify fractions. P(A') = 1 − P(A) — use this whenever it's easier to find the complement.
Identify mutually exclusive events (cannot both occur). Use the addition rule: P(A or B) = P(A) + P(B) for mutually exclusive events.
Construct sample space diagrams, lists, and tables to systematically list all possible outcomes of combined events.
Exam tip: For two dice or two spinners, draw a grid. Count systematically — it's easy to miss outcomes or double-count without a structured approach.
Draw and use tree diagrams for two or more events. Multiply along branches for AND; add separate branches for OR. Distinguish between independent (with replacement) and dependent (without replacement) events.
Exam tip: Without replacement: the second-event probabilities change after the first. The denominators decrease by 1. Always label each branch and check that probabilities on branches from the same node sum to 1.
Calculate conditional probabilities P(A|B) = P(A and B) / P(B). Understand what "given that" means and adjust the sample space accordingly.
Exam tip: P(A|B) means "probability of A, given B has happened." The denominator is now restricted to outcomes where B occurred. Venn diagrams help visualise the restricted sample space.
Construct and interpret Venn diagrams with up to three overlapping sets. Find probabilities of intersections, unions, and complements from Venn diagrams.
Exam tip: Fill in the Venn diagram from the inside out: start with the three-way intersection, then the pairwise intersections, then the outer regions. Always verify the total matches the given number.
Calculate mean, median, mode, and range from a list of data and from a frequency table (including finding the mean from a frequency table using Σfx ÷ Σf).
Exam tip: For median from a frequency table: find the (n+1)/2 th value using cumulative frequency. For mode from a table: it's the value with the highest frequency, not the highest value.
Read and interpret bar charts, pie charts, pictograms, line graphs, and stem-and-leaf diagrams. Draw and label charts accurately.
Exam tip: For pie charts: angle = (frequency ÷ total) × 360°. Back-calculate frequency from an angle: frequency = (angle ÷ 360°) × total.
Draw scatter diagrams. Describe correlation (positive, negative, none) and strength. Draw a line of best fit through the mean point. Use the line of best fit to estimate values and understand extrapolation risk.
Exam tip: Line of best fit must pass through the mean point (x̄, ȳ) — calculate this first. The line does not need to pass through any data points. Never extrapolate beyond the data range in exam answers.
Estimate the mean of grouped frequency data by using class midpoints: mean ≈ Σ(midpoint × frequency) ÷ Σfrequency.
Exam tip: Midpoint = (lower boundary + upper boundary) ÷ 2. The result is an estimate because original values are unknown. Show the midpoint × frequency column clearly in working.
Draw and interpret histograms using frequency density on the y-axis: frequency density = frequency ÷ class width. Area of bar = frequency.
Exam tip: The y-axis is ALWAYS frequency density — not frequency. Area = frequency. If the class width doubles and frequency stays the same, the bar height halves. Read class boundaries carefully — they are often non-integer.
Draw cumulative frequency curves (ogives) by plotting cumulative frequency against upper class boundaries. Read off the median, lower quartile (Q1), and upper quartile (Q3).
Exam tip: Plot points at the UPPER boundary of each class, starting from (lower boundary of first class, 0). Read Q1 at ¼n, median at ½n, Q3 at ¾n — always from the curve to the x-axis.
Calculate the interquartile range (IQR = Q3 − Q1) from a cumulative frequency curve or data set.
Exam tip: IQR is a measure of spread: it covers the middle 50% of the data. Read Q1 and Q3 from the cumulative frequency curve, then subtract. Compare IQRs across datasets to compare consistency.
Disclaimer: Based on the current Cambridge IGCSE Mathematics (0580) syllabus. Topic coverage, phrasing, and assessment objectives may vary between syllabus editions. Always verify content against the official Cambridge specification at cambridgeinternational.org before using for exam preparation.
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