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IGCSE Mathematics Formula Sheet (0580)
IGCSE Mathematics Formula Sheet
Cambridge IGCSE Mathematics (0580) | Essential Formulas
Based on official Cambridge syllabus 2025-2027
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Featured Formulas (8 key)
Across algebra, geometry, trig & coordinate geometry#1x = (−b ± √(b² − 4ac)) / 2a
Quadratic formula (Extended)
For ax² + bx + c = 0.
#2a² + b² = c²
Pythagoras' theorem
For a right-angled triangle: c is the hypotenuse (opposite the right angle), a and b are the other two sides.
#3sin θ = O/H · cos θ = A/H · tan θ = O/A
SOH-CAH-TOA (right-angled trigonometry)
O = Opposite
A = Adjacent
H = Hypotenuse
A = Adjacent
H = Hypotenuse
#4a / sin A = b / sin B = c / sin C
Sine rule (non-right-angled triangles) (Extended)
Use when you have an angle paired with its opposite side. Each side is opposite the angle it shares a letter with.
#5y = mx + c
Equation of a straight line (gradient-intercept form)
m = Gradient (slope)
c = y-intercept (where the line crosses the y-axis)
c = y-intercept (where the line crosses the y-axis)
#6m = (y₂ − y₁) / (x₂ − x₁)
Gradient between two points (Extended)
Rise over run. Positive m = uphill, negative = downhill, 0 = horizontal.
#7Area = π × r²
Area of a circle
r = Radius. Circumference uses the same r: C = 2πr.
#8Percentage = (Part / Whole) × 100
Percentage calculation
For % change: (New − Old) / Old × 100. Negative means decrease.
Given on the Exam Paper
Cambridge prints these on page 2 of each paper. Learn the rest of this sheet for your tier (Core: skip items marked Extended), except the density, pressure and population density formulas, which are given in the question.
Core and Extended (Papers 1, 2, 3 and 4)
- Area of a triangle: A = ½bh
- Area of a circle: A = πr²
- Circumference of a circle: C = 2πr
- Curved surface area of a cylinder: A = 2πrh
- Curved surface area of a cone: A = πrl (l = sloping edge)
- Surface area of a sphere: A = 4πr²
- Volume of a prism: V = Al (A = cross-sectional area)
- Volume of a pyramid: V = ⅓Ah (A = base area)
- Volume of a cylinder: V = πr²h
- Volume of a cone: V = ⅓πr²h
- Volume of a sphere: V = (4/3)πr³
Also on Extended papers only (Papers 2 and 4)
- For ax² + bx + c = 0, where a ≠ 0: x = (−b ± √(b² − 4ac)) / 2a
- Sine rule: a/sin A = b/sin B = c/sin C
- Cosine rule: a² = b² + c² − 2bc cos A
- Area of a triangle: Area = ½ab sin C
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1. Number
Percentage = (Part/Whole) × 100
Writing one amount as a percentage of another
Percentage of a quantity = (Percentage/100) × Quantity
Finding a percentage of a value
% Change = (Change/Original) × 100
Works for increase or decrease
Reverse % (Extended): Original = Final / (1 ± r/100)
r = percentage rate; + for increase, − for decrease
Simple Interest: I = PRT/100
P = principal, R = rate (%), T = time (years)
Compound Interest: A = P(1 + r/100)ⁿ
A = final amount, P = starting amount, r = percentage rate, n = number of periods
Exponential growth and decay (Extended):
A = P(1 ± r/100)ⁿ
A = P(1 ± r/100)ⁿ
r = percentage rate; + for growth (a rising population), − for decay (a car losing value)
Speed = Distance / Time
D = S × T, T = D / S
Density = Mass / Volume
M = D × V, V = M / D (the formula is given in the question)
Standard form: A × 10ⁿ
1 ≤ A < 10, n = positive or negative integer
Surds (Extended): √(ab) = √a × √b
Simplify with a square factor: √18 = √9 × √2 = 3√2
Rationalising the denominator (Extended):
a/√b = a√b / b
a/√b = a√b / b
Multiply top and bottom by √b. For 1/(a + √b), multiply top and bottom by a − √b
Prime factors: 60 = 2² × 3 × 5
HCF = highest common factor, LCM = lowest common multiple
Reciprocal of x = 1/x
Rational = can be written as a fraction. Irrational = cannot, such as √2 and π. Know natural numbers, integers, primes, squares and cubes
Recall: squares from 1² to 15²
Also cubes of 1, 2, 3, 4, 5 and 10, and the matching roots
Sets: n(A), A′, A ∪ B, A ∩ B
Number of elements, complement, union, intersection. ℰ = universal set. Venn diagrams: two sets
Sets (Extended): ∈, ∉, ∅, A ⊆ B, A ⊈ B
Element of, not an element of, empty set, subset, not a subset. Venn diagrams: up to three sets
Recurring decimals (Extended): 0.777… = 7/9
x = 0.777…, so 10x − x = 7. Written with a dot over the repeating digit
Order of operations: brackets, indices, × and ÷, + and −
Also with negatives, fractions and mixed numbers. Convert between fractions, decimals and percentages
Accuracy: non-exact answers 3 significant figures, angles 1 decimal place
Unless the question says otherwise. Round only the final answer. Estimate by rounding each number to 1 significant figure. π = calculator value or 3.142
Calculator display: read it in context
In money 2.7 means $2.70. In time 1.5 means 1 hour 30 minutes. Enter 1 hour 45 minutes as 1.75
Bounds: value ± half the rounding unit
7 m to the nearest metre: 6.5 ≤ length < 7.5. Calculations (Extended): upper bound of a ÷ b = upper a ÷ lower b
Ratio a : b: share = a/(a + b) × total
Simplify by dividing by a common factor. Use for recipes, map scales and best value
Rates: amount = rate × quantity
For pay, currency, flow and fuel use. Pressure and population density formulas are given in the question
Time: 1 year = 365 days
24-hour clock: 6.40 p.m. = 18 40. Includes time zones and timetables
2. Algebra
Quadratic Formula (Extended):
x = (-b ± √(b²-4ac)) / 2a
x = (-b ± √(b²-4ac)) / 2a
For ax² + bx + c = 0
Difference of squares (Extended):
a² - b² = (a+b)(a-b)
a² - b² = (a+b)(a-b)
Useful for factorising
(a + b)² = a² + 2ab + b²
Perfect square expansion
(a - b)² = a² - 2ab + b²
Perfect square expansion
nth term (linear): aₙ = a + (n-1)d
a = first term, d = common difference
nth term (quadratic) (Extended): aₙ = an² + bn + c
Second difference ÷ 2 = a. Core covers simple quadratics only
Completing the square (Extended):
x² + bx + c = (x + b/2)² − (b/2)² + c
x² + bx + c = (x + b/2)² − (b/2)² + c
For ax² + bx + c, take out a first. The turning point of y = (x + p)² + q is (−p, q)
Direct proportion (Extended):
y ∝ x → y = kx
y ∝ x → y = kx
k = constant. x can be replaced by x², x³, √x or ∛x
Inverse proportion (Extended):
y ∝ 1/x → y = k/x
y ∝ 1/x → y = k/x
k = constant. x can be replaced by x², x³, √x or ∛x
Speed-time graph (Extended):
acceleration = gradient
acceleration = gradient
Negative gradient = deceleration. Distance = area under the graph (straight sections only)
Differentiation (Extended):
y = axⁿ → dy/dx = naxⁿ⁻¹
y = axⁿ → dy/dx = naxⁿ⁻¹
n = positive integer or zero. For a sum, differentiate each term. dy/dx gives the gradient
Turning points (Extended): dy/dx = 0
d²y/dx² positive = minimum, negative = maximum. Or check the gradient either side
Gradient of a curve (Extended): draw a tangent
Gradient of the tangent = gradient of the curve at that point
Functions (Extended): gf(x) = g(f(x))
Apply f first. For f⁻¹(x), swap x and y in y = f(x), then make y the subject. Domain = input values, range = output values
Common factor: 6x² + 9xy = 3x(2x + 3y)
Substitute numbers for letters, collect like terms, expand brackets, factorise fully
Factorising (Extended):
ax + bx + kay + kby = (a + b)(x + ky)
ax + bx + kay + kby = (a + b)(x + ky)
Also factorise ax² + bx + c and ax³ + bx² + cx. Quadratics can be solved by factorising
Algebraic fractions (Extended):
a/b + c/d = (ad + bc)/bd
a/b + c/d = (ad + bc)/bd
Factorise first, then cancel common factors
Equations: form them from the words, then do the same to both sides
Fractional equations (Extended): clear the denominators first
Simultaneous equations: eliminate or substitute
One linear, one non-linear (Extended): substitute the linear one into the other
Change the subject: undo one step at a time
Subject appears twice (Extended): collect it, then factorise. Power or root of the subject (Extended): undo it
Inequalities: open circle for < or >, closed circle for ≤ or ≥
On a number line. Also order quantities with = ≠ > < ≥ ≤
Inequality regions (Extended): broken line for < or >, solid line for ≤ or ≥
Shade the unwanted region unless told otherwise. List the inequalities that define a region. Solve linear inequalities like equations, but multiplying or dividing by a negative number reverses the inequality sign
Other sequences: compare with n³
Simple cubic. Term-to-term rule = how to get the next term. Exponential (Extended): compare with powers such as 2ⁿ
Distance-time graph: gradient = speed
On straight-line graphs, gradient = rate of change. Read travel and conversion graphs from the axes
Graphs: y = ax + b, y = ±x² + ax + b, y = a/x
Use a table of values. Solve equations from where graphs cross. Sketches show roots and symmetry
More graphs (Extended): y = axⁿ, y = abˣ + c
n = −2, −1, −½, 0, ½, 1, 2, 3. Also sketch cubic, reciprocal and exponential curves. y = a/x + b has asymptotes x = 0 and y = b. y = arˣ + b has asymptote y = b
3. Indices (Powers)
aᵐ × aⁿ = aᵐ⁺ⁿ
aᵐ ÷ aⁿ = aᵐ⁻ⁿ
(aᵐ)ⁿ = aᵐⁿ
a⁰ = 1
a⁻ⁿ = 1/aⁿ
a^(1/n) = ⁿ√a (Extended)
a^(m/n) = ⁿ√(aᵐ) (Extended)
(ab)ⁿ = aⁿbⁿ
Solve 2ˣ = 8: 2ˣ = 2³, so x = 3
4. Coordinate Geometry
Gradient (Extended): m = (y₂-y₁)/(x₂-x₁)
Rise over run between two points. Core finds gradient from a grid only
Straight line: y = mx + c
m = gradient, c = y-intercept
Midpoint (Extended): ((x₁+x₂)/2, (y₁+y₂)/2)
Average of coordinates
Distance (Extended): d = √[(x₂-x₁)² + (y₂-y₁)²]
Pythagoras between two points
Parallel lines: m₁ = m₂
Same gradient
Perpendicular (Extended): m₁ × m₂ = -1
Gradients are negative reciprocals
Coordinates (x, y); vertical line: x = k
Across, then up. Plot points and rule the line. ax + by = c (Extended): rearrange to y = mx + c for the gradient
5. Areas
Rectangle
A = l × w
Triangle
A = ½ × b × h
Parallelogram
A = b × h
Trapezium
A = ½(a + b) × h
Circle
A = πr²
Sector
A = (θ/360) × πr²
Triangle (with angle) (Extended)
A = ½ × a × b × sin(C)
Two sides and included angle
6. Perimeter & Circumference
Rectangle
P = 2(l + w)
Circle
C = 2πr = πd
Arc length
l = (θ/360) × 2πr
Sector perimeter
P = arc + 2r
7. Volume
Cuboid
V = l × w × h
Prism
V = cross-sectional area × length
Cylinder
V = πr²h
Cone
V = ⅓πr²h
Sphere
V = (4/3)πr³
Pyramid
V = ⅓ × base area × h
Compound shapes and solids
Split into known parts, then add or subtract
Half a sphere = ½ × whole sphere. Frustum (Extended) = large cone − small cone
8. Surface Area
Cuboid
SA = 2(lw + lh + wh)
Cylinder
SA = 2πr² + 2πrh
Sphere
SA = 4πr²
Cone (curved)
CSA = πrl
l = slant height
Unit conversions
1 km = 1000 m, 1 m = 100 cm, 1 cm = 10 mm
1 km² = 1 000 000 m², 1 m² = 10 000 cm², 1 cm² = 100 mm²
1 m³ = 1 000 000 cm³, 1 cm³ = 1000 mm³
1 m³ = 1000 l, 1 l = 1000 ml, 1 kg = 1000 g
1 km² = 1 000 000 m², 1 m² = 10 000 cm², 1 cm² = 100 mm²
1 m³ = 1 000 000 cm³, 1 cm³ = 1000 mm³
1 m³ = 1000 l, 1 l = 1000 ml, 1 kg = 1000 g
9. Pythagoras & Trigonometry
Pythagoras' Theorem
a² + b² = c²
c = hypotenuse (longest side, opposite right angle)
SOH
sin θ = O/H
CAH
cos θ = A/H
TOA
tan θ = O/A
Sine Rule (Extended)
a/sin A = b/sin B = c/sin C
Cosine Rule (Extended)
a² = b² + c² - 2bc cos A
Use Sine Rule when: 2 angles + 1 side, or 2 sides + angle opposite one of them
Use Cosine Rule when: 3 sides, or 2 sides + included angle
Use Cosine Rule when: 3 sides, or 2 sides + included angle
Exact values (Extended)
sin
0° = 0
30° = ½
45° = √2/2
60° = √3/2
90° = 1
30° = ½
45° = √2/2
60° = √3/2
90° = 1
cos
0° = 1
30° = √3/2
45° = √2/2
60° = ½
90° = 0
30° = √3/2
45° = √2/2
60° = ½
90° = 0
tan
0° = 0
30° = √3/3
45° = 1
60° = √3
30° = √3/3
45° = 1
60° = √3
Ambiguous case of the sine rule (Extended)
sin x = sin(180° − x)
Two sides and an angle opposite one of them can give an acute or an obtuse answer
Trig graphs and equations (Extended)
y = sin x, y = cos x, y = tan x
For 0° ≤ x ≤ 360°. Find other solutions from the graph: cos x = cos(360° − x), tan x = tan(180° + x)
Elevation and depression (Extended)
Angle up or down from the horizontal
Shortest distance from a point to a line = perpendicular distance
3D problems (Extended)
d² = a² + b² + c²
Diagonal of a cuboid. For the angle between a line and a plane, use the right-angled triangle that contains it
10. Angles & Polygons
Angles in triangle = 180°
Angles in quadrilateral = 360°
Angles on straight line = 180°
Angles at a point = 360°
Vertically opposite angles are equal
Parallel lines: corresponding angles are equal
Parallel lines: alternate angles are equal
Parallel lines: co-interior angles add to 180°
Interior angle sum (n-gon): (n - 2) × 180°
Each interior angle (regular): (n - 2) × 180° / n
Each exterior angle (regular): 360° / n
Exterior angles of a polygon add to: 360°
Polygon names: pentagon 5 sides, hexagon 6, octagon 8, decagon 10
Regular polygon with n sides: n lines of symmetry, rotational order n
Symmetry of solids (Extended): planes and axes of prisms, cylinders, pyramids, cones
Three-letter notation: angle ABC = the angle at B
Terms to know: triangle and quadrilateral types, acute, obtuse and reflex angles, solids, faces and edges
Circle theorems
Circle words: radius, diameter, chord, tangent, arc, sector, segment. Major and minor arc (Extended)
Angle in a semicircle = 90°
Tangent meets radius at 90°
On the same arc, angle at centre = 2 × angle at circumference (Extended)
Angles in the same segment are equal (Extended)
Opposite angles of a cyclic quadrilateral add to 180° (Extended)
Alternate segment theorem: angle between tangent and chord = angle in the alternate segment (Extended)
Equal chords are the same distance from the centre (Extended)
Perpendicular bisector of a chord goes through the centre (Extended)
Two tangents from one outside point are equal in length (Extended)
Bearings: three figures, clockwise from north
Reverse bearing = bearing ± 180°
Drawing: triangle from three sides with ruler and compasses, keeping the arcs
Also scale drawings and nets
11. Similarity & Enlargement
Scale Factor (SF): new length ÷ original length
Similar shapes - Length: ratio = k
Similar shapes - Area (Extended): ratio = k²
Similar shapes - Volume (Extended): ratio = k³
If lengths are in ratio 1:2, areas are in ratio 1:4, volumes are in ratio 1:8 (Extended)
Similar or congruent: similar = same shape, congruent = same shape and size
Showing two triangles are similar (Extended): give reasons why matching angles are equal
Transformations: reflection (mirror line), rotation (centre, angle, direction), enlargement (centre, scale factor), translation (vector)
Negative scale factors and combined transformations (Extended)
12. Statistics & Probability
Mean: Σx / n
Frequency table: Σfx / Σf
Estimate of mean, grouped data (Extended): Σfx / Σf
x = class midpoint
Median: middle value of ordered data; mean of the middle two if the count is even
Mode: most frequent value
Range: highest − lowest
Interquartile range (Extended): upper quartile − lower quartile
(quartiles sit ¼ and ¾ of the way through ordered data)
Frequency density (Extended): frequency ÷ class width
(the vertical axis of a histogram; bar area = frequency)
Modal class (Extended): class with the highest frequency
Cumulative frequency (Extended): running total, plotted with a smooth curve
Read the median, quartiles and percentiles from the curve
Pie chart angle: frequency ÷ total × 360°
Also tally and two-way tables, bar charts, pictograms, frequency distributions, and stem-and-leaf diagrams (ordered, with a key)
Scatter diagrams: positive, negative or zero correlation
Line of best fit: one ruled line, with points roughly even either side
Comparing data: compare an average and the range
Interquartile range too (Extended). Conclusions from data have limits
Probability: P(A) = favourable outcomes / total outcomes
Probabilities run from 0 to 1
P(A and B): P(A) × P(B)
(independent)
P(A or B): P(A) + P(B)
(mutually exclusive)
P(not A): 1 - P(A)
Written P(A′) (Extended)
Expected frequency: probability × number of trials
Relative frequency: times it happened ÷ number of trials
An estimate of probability. Fair = equally likely outcomes, bias = not equally likely, random = left to chance
Combined events: sample space, Venn and tree diagrams
Tree diagrams: multiply along branches, add the routes. Core uses with replacement only. Without replacement (Extended): second-stage probabilities change
Conditional probability (Extended): count only the outcomes in the given group
Use a Venn diagram, tree diagram or table. No formula needed
13. Vectors
Magnitude (Extended): |a| = √(x² + y²)
x and y = the two numbers in the column vector a, drawn as a directed line segment
Add or subtract (Extended): top with top, bottom with bottom
Multiply by a scalar (Extended): multiply both numbers by the scalar
Position vectors (Extended): vector AB = b − a
a and b = position vectors of A and B
Parallel vectors (Extended): one is a multiple of the other
Collinear points: parallel vectors that share a point
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© LumiExams.com | IGCSE Mathematics Formula Sheet | Based on Cambridge IGCSE Mathematics 0580 Syllabus (2025-2027)
Frequently Asked Questions
What formulas are given in IGCSE Maths exams?
Cambridge prints a List of formulas on page 2 of every paper. Core papers (Paper 1 and Paper 3) and Extended papers (Paper 2 and Paper 4) both give: area of a triangle (A = ½bh), area of a circle (A = πr²), circumference of a circle (C = 2πr), curved surface area of a cylinder (A = 2πrh), curved surface area of a cone (A = πrl, where l is the sloping edge), surface area of a sphere (A = 4πr²), and the volumes of a prism (V = Al, where A is the cross-sectional area), pyramid (V = ⅓Ah, where A is the base area), cylinder (V = πr²h), cone (V = ⅓πr²h) and sphere (V = (4/3)πr³). Extended papers also give the quadratic formula for ax² + bx + c = 0 where a ≠ 0, the sine rule, the cosine rule (a² = b² + c² − 2bc cos A) and Area = ½ab sin C. However, you must memorize other formulas, such as Pythagoras theorem, SOH-CAH-TOA, and simple/compound interest.
What formulas should I memorize for IGCSE Maths?
Memorize: gradient formula (Extended), distance formula (Extended), circle theorems, area formulas (rectangle, parallelogram, trapezium), Pythagoras theorem (a²+b²=c²), SOH-CAH-TOA for trigonometry, and compound interest formula.
What topics are covered in IGCSE Mathematics?
IGCSE Mathematics (0580) covers: Number, Algebra & Graphs, Coordinate Geometry, Geometry, Mensuration (area/volume), Trigonometry, Vectors & Transformations, Probability, and Statistics. Our formula sheet organizes key formulas by topic.
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