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Official STEM Exam Formula Handbooks

Quick reference sheets with authentic KaTeX-rendered LaTeX notation used during official Cambridge, AP, and IB science and mathematics assessments. Always accessible in the digital exam hall.

Cambridge Assessment

Cambridge IGCSE Mathematics (0580)

IGCSE Mathematics (Extended)

Practice this Syllabus
Curved Surface Area of a Cone
A=πrlA = \pi r l
where r is base radius and l is slant height
Surface Area of a Sphere
A=4πr2A = 4 \pi r^2
total surface area of a 3D sphere
Volume of a Cone
V=13πr2hV = \frac{1}{3} \pi r^2 h
where h is the vertical perpendicular height
Volume of a Sphere
V=43πr3V = \frac{4}{3} \pi r^3
solid volume calculation
Cosine Rule for Any Triangle
a2=b2+c2−2bccos⁡Aa^2 = b^2 + c^2 - 2 b c \cos A
or \cos A = \frac{b^2 + c^2 - a^2}{2bc}
Area of Triangle with Angle
Area=12absin⁡C\text{Area} = \frac{1}{2} a b \sin C
included angle C between sides a and b
Cambridge Assessment

Cambridge IGCSE International Mathematics (0607)

Paper 1 (Core) Official Formula List

Practice this Syllabus
Area of Triangle & Parallelogram
A=12bh;A=bhA = \frac{1}{2}bh \quad ; \quad A = bh
Base b, perpendicular height h
Area of Trapezium
A=12(a+b)hA = \frac{1}{2}(a + b)h
Parallel sides a, b with perpendicular separation h
Curved Surface Area of Cone
A=πrlA = \pi r l
Radius r, sloping edge l
Volume of Prism & Cylinder
V=A×l;V=πr2hV = A \times l \quad ; \quad V = \pi r^2 h
Cross-sectional area A and prism length l
Volume of Sphere & Cone
V=43πr3;V=13πr2hV = \frac{4}{3}\pi r^3 \quad ; \quad V = \frac{1}{3}\pi r^2 h
Curved solids volume formulas
College Board AP

AP Calculus BC Formula Sheet

AP Calculus BC

Practice this Syllabus
General Taylor Series centered at x = a
f(x)=∑n=0∞f(n)(a)n!(x−a)nf(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n
Power series expansion centered at a
e^x Maclaurin Series
ex=∑n=0∞xnn!=1+x+x22!+x33!+⋯e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots
Radius of convergence R = \infty
sin(x) Maclaurin Series
sin⁡x=∑n=0∞(−1)nx2n+1(2n+1)!=x−x33!+x55!−⋯\sin x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots
Odd symmetric powers
Polar Area Integral
Area=12∫αβ[r(θ)]2 dθ\text{Area} = \frac{1}{2} \int_{\alpha}^{\beta} [r(\theta)]^2 \, d\theta
Continuous radial boundary region
Parametric Arc Length
L=∫t1t2(dxdt)2+(dydt)2 dtL = \int_{t_1}^{t_2} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt
Total trajectory curve path length
Integration by Parts
∫u dv=uv−∫v du\int u \, dv = u v - \int v \, du
L-I-A-T-E priority rule
IB Diploma Programme

IB DP Mathematics: Analysis & Approaches (HL)

Mathematics AA (HL)

Practice this Syllabus
Euler’s & Polar Form of Complex Numbers
z=r(cos⁡θ+isin⁡θ)=reiθz = r(\cos \theta + i \sin \theta) = r e^{i\theta}
r = |z|, \theta = \operatorname{Arg}(z) in radians
De Moivre’s Theorem
[r(cos⁡θ+isin⁡θ)]n=rn(cos⁡nθ+isin⁡nθ)[r(\cos \theta + i \sin \theta)]^n = r^n (\cos n\theta + i \sin n\theta)
Valid for integer and rational powers n
Integrating Factor (Linear 1st Order ODE)
I(x)=e∫P(x) dxI(x) = e^{\int P(x) \, dx}
For differential equation dy/dx + P(x)y = Q(x)
Infinite Geometric Series Sum
S∞=u11−rS_\infty = \frac{u_1}{1 - r}
Converges when |r| < 1
Cambridge Assessment

Cambridge International A Level Mathematics (9709)

A Level Mathematics (MF19 Tables)

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Binomial Expansion for Any Index
(1+x)n=1+nx+n(n−1)2!x2+⋯(1 + x)^n = 1 + n x + \frac{n(n-1)}{2!} x^2 + \cdots
Converges for |x| < 1 when n is rational or negative
Arithmetic Series Sum
Sn=n2[2a+(n−1)d]S_n = \frac{n}{2}[2a + (n-1)d]
n terms with first term a and common difference d
Double Angle Sine Identity
sin⁡2A=2sin⁡Acos⁡A\sin 2A = 2 \sin A \cos A
Fundamental trigonometric double angle formulation
Second Derivative Nature Test
d2ydx2>0  ⟹  Min;d2ydx2<0  ⟹  Max\frac{d^2y}{dx^2} > 0 \implies \text{Min} \quad ; \quad \frac{d^2y}{dx^2} < 0 \implies \text{Max}
Stationary point curvature classification
College Board AP

AP Physics 1: Mechanics & Dynamics

AP Physics 1 (Algebra-Based)

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1D Kinematics with Constant Acceleration
v=v0+at;x=x0+v0t+12at2v = v_0 + at \quad ; \quad x = x_0 + v_0 t + \frac{1}{2} a t^2
Linear motion under constant force
Torque & Rotational Equilibrium
τ=rFsin⁡θ=Iα\tau = r F \sin \theta = I \alpha
Rotational analog of Newton's Second Law
Mechanical Energy & Work
Ek=12mv2;Ug=mgh;W=Fdcos⁡θE_k = \frac{1}{2}mv^2 \quad ; \quad U_g = mgh \quad ; \quad W = F d \cos\theta
Conservation of mechanical energy
Simple Harmonic Period (Pendulum & Spring)
Ts=2πmk;Tp=2πLgT_s = 2\pi \sqrt{\frac{m}{k}} \quad ; \quad T_p = 2\pi \sqrt{\frac{L}{g}}
Oscillation period equations
International Science & AP/IB Chemistry

STEM Chemistry & Thermochemistry Constants

Chemical Energetics & Reaction Kinetics

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Gibbs Free Energy Equation
ΔG∘=ΔH∘−TΔS∘=−RTln⁡K\Delta G^\circ = \Delta H^\circ - T\Delta S^\circ = -RT \ln K
Spontaneity criteria: spontaneous when ΔG < 0
Ideal Gas Law with Universal Gas Constant
PV=nRT(R=8.314 J⋅mol−1⋅K−1)PV = nRT \quad (R = 8.314 \text{ J}\cdot\text{mol}^{-1}\cdot\text{K}^{-1})
Equation of state for ideal gases
Arrhenius Reaction Rate Constant
k=Ae−EaRTorln⁡k=−EaR(1T)+ln⁡Ak = A e^{-\frac{E_a}{RT}} \quad \text{or} \quad \ln k = -\frac{E_a}{R}\left(\frac{1}{T}\right) + \ln A
Temperature dependence of reaction rate constant
Henderson-Hasselbalch Buffer Equation
pH=pKa+log⁡([A−][HA])\text{pH} = \text{p}K_a + \log\left(\frac{[\text{A}^-]}{[\text{HA}]}\right)
Buffer solution pH prediction and acid dissociation