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IGCSE Mathematics Formula Sheet (0580)
IGCSE Mathematics Formula Sheet
Cambridge IGCSE Mathematics (0580) | All Essential Formulas
Based on official Cambridge syllabus 2023-2025
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Featured Formulas (8 core)
Across algebra, geometry, trig & coordinate geometry#1x = (−b ± √(b² − 4ac)) / 2a
Quadratic formula
For ax² + bx + c = 0. Discriminant b² − 4ac tells you the number of real roots: >0 two roots, =0 one, <0 none.
#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 = O/H · cos = A/H · tan = 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)
Use when you have an angle paired with its opposite side. Each side is opposite the angle it shares a letter with.
#5y = m × x + 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
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.
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1. Number
Percentage = (Part/Whole) × 100
Finding a percentage of a value
% Change = (Change/Original) × 100
Works for increase or decrease
Reverse % : Original = Final / (1 ± r)
r = percentage as decimal; + 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, n = number of periods
Speed = Distance / Time
D = S × T, T = D / S
Density = Mass / Volume
M = D × V, V = M / D
2. Algebra
Quadratic Formula:
x = (-b ± √(b²-4ac)) / 2a
x = (-b ± √(b²-4ac)) / 2a
For ax² + bx + c = 0
Discriminant: b² - 4ac (>0: 2 roots, =0: 1 root, <0: no real roots)
Discriminant: b² - 4ac (>0: 2 roots, =0: 1 root, <0: no real roots)
Difference of squares:
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): aₙ = an² + bn + c
Second difference ÷ 2 = a
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
a^(m/n) = ⁿ√(aᵐ)
(ab)ⁿ = aⁿbⁿ
4. Coordinate Geometry
Gradient: m = (y₂-y₁)/(x₂-x₁)
Rise over run between two points
Straight line: y = mx + c
m = gradient, c = y-intercept
Midpoint: ((x₁+x₂)/2, (y₁+y₂)/2)
Average of coordinates
Distance: d = √[(x₂-x₁)² + (y₂-y₁)²]
Pythagoras between two points
Parallel lines: m₁ = m₂
Same gradient
Perpendicular: m₁ × m₂ = -1
Gradients are negative reciprocals
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)
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 = Area × length
Cylinder
V = πr²h
Cone
V = ⅓πr²h
Sphere
V = (4/3)πr³
Pyramid
V = ⅓ × base area × h
8. Surface Area
Cuboid
SA = 2(lw + lh + wh)
Cylinder
SA = 2πr² + 2πrh
Sphere
SA = 4πr²
Cone (curved)
CSA = πrl
l = slant height
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
a/sin A = b/sin B = c/sin C
Cosine Rule
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
10. Angles & Polygons
Angles in triangle = 180°
Angles in quadrilateral = 360°
Angles on straight line = 180°
Angles at a point = 360°
Interior angle sum (n-gon): (n - 2) × 180°
Each interior angle (regular): (n - 2) × 180° / n
Each exterior angle (regular): 360° / n
11. Similarity & Enlargement
Scale Factor (SF): new length ÷ original length
Similar shapes - Length: ratio = k
Similar shapes - Area: ratio = k²
Similar shapes - Volume: ratio = k³
If lengths are in ratio 1:2, areas are in ratio 1:4, volumes are in ratio 1:8
12. Statistics & Probability
Mean: Σx / n
Grouped Mean: Σfx / Σf
Probability: P(A) = favourable outcomes / total outcomes
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)
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