Maths Formula Sheet
Every formula a school or first-year university maths student needs, organised by topic. Bookmark this page.
Algebra
Quadratic formula
x = (−b ± √(b² − 4ac)) / (2a)
Roots of ax² + bx + c = 0.
Difference of squares
a² − b² = (a + b)(a − b)
Factorise any expression of this form.
Sum of cubes
a³ + b³ = (a + b)(a² − ab + b²)
Difference of cubes
a³ − b³ = (a − b)(a² + ab + b²)
Binomial expansion
(a + b)ⁿ = Σ C(n, k) a^(n−k) b^k
C(n, k) = n! / (k!(n−k)!).
Arithmetic progression sum
Sₙ = n/2 · (a₁ + aₙ)
Geometric progression sum
Sₙ = a₁ · (1 − rⁿ) / (1 − r)
For r ≠ 1.
Logarithm rules
log(ab) = log a + log b, log(aⁿ) = n log a
Exponential rules
a^m · a^n = a^(m+n), (a^m)^n = a^(mn)
Geometry
Pythagoras
a² + b² = c²
For the sides of a right triangle.
Triangle area (Heron)
√(s(s−a)(s−b)(s−c))
s = (a + b + c)/2.
Circle area
πr²
Circle circumference
2πr
Sphere volume
4/3 · πr³
Sphere surface area
4πr²
Cylinder volume
πr²h
Cone volume
1/3 · πr²h
Cuboid surface area
2(lw + lh + wh)
Trigonometry
Pythagorean identity
sin²θ + cos²θ = 1
Sine of sum
sin(A + B) = sin A cos B + cos A sin B
Cosine of sum
cos(A + B) = cos A cos B − sin A sin B
Tangent of sum
tan(A + B) = (tan A + tan B) / (1 − tan A tan B)
Double angle (sin)
sin(2A) = 2 sin A cos A
Double angle (cos)
cos(2A) = cos²A − sin²A
Law of sines
a/sin A = b/sin B = c/sin C
Law of cosines
c² = a² + b² − 2ab cos C
Calculus
Power rule
d/dx (xⁿ) = n · x^(n−1)
Product rule
(fg)′ = f′g + fg′
Quotient rule
(f/g)′ = (f′g − fg′) / g²
Chain rule
(f ∘ g)′(x) = f′(g(x)) · g′(x)
Derivative of sin
d/dx sin x = cos x
Derivative of e^x
d/dx e^x = e^x
Derivative of ln
d/dx ln x = 1 / x
Integral of xⁿ
∫ xⁿ dx = x^(n+1) / (n + 1) + C
For n ≠ −1.
Integral of 1/x
∫ 1/x dx = ln|x| + C
Integral of e^x
∫ e^x dx = e^x + C
Fundamental theorem
∫ₐᵇ f′(x) dx = f(b) − f(a)
Statistics & probability
Mean
μ = (1/n) Σ xᵢ
Sample variance
s² = (1/(n−1)) Σ (xᵢ − x̄)²
Standard deviation
σ = √(σ²)
Z-score
z = (x − μ) / σ
Bayes's theorem
P(A|B) = P(B|A) P(A) / P(B)
Combinations
C(n, k) = n! / (k! (n−k)!)
Permutations
P(n, k) = n! / (n−k)!
Numbers
Greatest common divisor
gcd(a, b) = gcd(b, a mod b)
Least common multiple
lcm(a, b) = |ab| / gcd(a, b)
Number of divisors
τ(n) = ∏ (aᵢ + 1)
Where n = ∏ pᵢ^aᵢ.