STEM Formula Cheat Sheet | Physics, Chemistry, Biology, Mathematics

Best Strategy for Using This Cheat Sheet

Identify the subject: Start by choosing the correct category—physics, chemistry, mathematics, or biology—so you use the right equation set.
Understand the variables: Write down what each symbol means and keep units consistent before plugging in numbers.
Choose the correct formula: Match the problem’s known values to the formula that contains the unknown you need.
Rearrange carefully: If the unknown is not isolated, rearrange the formula step-by-step instead of guessing.
Check units and signs: Confirm SI units where possible, and pay attention to negative signs, angles, and direction-sensitive quantities.
Use step-by-step verification: After calculation, review each step and compare with a secondary formula if available.
Combine formulas only when needed: For multi-step problems, solve one variable at a time and then plug it into the next formula.

Physics

Core formulas and quick explanations for physics.

Newton’s Second Law

F = ma

Force equals mass times acceleration.

Weight

W = mg

Weight is mass times gravitational acceleration.

Work Done

W = Fd \cos\theta

Work is the force component along displacement.

Kinetic Energy

KE = \frac{1}{2}mv^2

Energy of a moving object based on mass and speed.

Potential Energy

PE = mgh

Gravitational potential energy near Earth’s surface.

Power

P = \frac{W}{t} = Fv

Energy per time or force times velocity.

Equations of Motion

v = u + at s = ut + \frac{1}{2}at^2 v^2 = u^2 + 2as

Core kinematics relationships for constant acceleration.

Momentum

p = mv

Mass times velocity gives linear momentum.

Impulse

J = Ft = \Delta p

Impulse equals change in momentum.

Pressure

P = \frac{F}{A}

Force spread over an area.

Fluid Pressure

P = \rho gh

Pressure due to a fluid column at depth.

Ohm’s Law

V = IR

Voltage equals current times resistance.

Electrical Power

P = VI = I^2R = \frac{V^2}{R}

Electrical power in circuits.

Charge

Q = It

Charge is current multiplied by time.

Wave Speed

v = f\lambda

Wave speed equals frequency times wavelength.

Snell’s Law

n_1 \sin\theta_1 = n_2 \sin\theta_2

Refraction law for light crossing media.

Refractive Index

n = \frac{c}{v}

Index of refraction relative to light speed in vacuum.

Lens Formula

\frac{1}{f} = \frac{1}{v} - \frac{1}{u}

Thin lens equation using object and image distances.

Magnification

m = -\frac{v}{u} = \frac{h'}{h}

Image size and orientation compared to object.

Gravitational Force

F = G\frac{m_1 m_2}{r^2}

Newton’s law of universal gravitation.

Coulomb’s Law

F = k\frac{q_1 q_2}{r^2}

Electrostatic force between point charges.

Frequency

f = \frac{1}{T}

The relationship between period and frequency.

Hooke's Law

F = -kx

Restoring force in a spring proportional to displacement.

Centripetal Force

F_c = \frac{mv^2}{r}

Force keeping an object in circular motion.

Torque

\tau = rF \sin\theta

Rotational force or moment of force.

Angular Momentum

L = I\omega

Rotational equivalent of linear momentum.

Efficiency

\eta = \frac{W_{out}}{W_{in}} \times 100\%

Ratio of useful work output to input.

Chemistry

Core formulas and quick explanations for chemistry.

Moles

n = \frac{m}{M}

Amount of substance from mass and molar mass.

Molar Concentration

c = \frac{n}{V}

Moles per unit volume.

Ideal Gas Law

PV = nRT

Pressure, volume, temperature, and amount of gas.

Boyle’s Law

P_1 V_1 = P_2 V_2

Pressure and volume for fixed temperature.

Charles’s Law

\frac{V_1}{T_1} = \frac{V_2}{T_2}

Volume is proportional to absolute temperature.

Gay-Lussac’s Law

\frac{P_1}{T_1} = \frac{P_2}{T_2}

Pressure is proportional to temperature at constant volume.

Combined Gas Law

\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}

Combines Boyle’s, Charles’s, and Gay-Lussac’s laws.

Avogadro Constant

N_A = 6.022 × 10^{23} \text{ mol}^{-1}

Particles per mole.

pH Definition

pH = -\log[H^+]

Measure of acidity.

pOH

pOH = -\log[OH^-]

Measure of basicity.

Acid Dissociation Constant

K_a = \frac{[H^+][A^-]}{[HA]}

Strength of a weak acid.

Dilution Law

C_1 V_1 = C_2 V_2

Moles stay constant during dilution.

Nernst Equation

E = E^\circ - \frac{RT}{nF} \ln Q

Cell potential for non-standard conditions.

Rate Law

r = k[A]^m[B]^n

Reaction rate dependence on reactant concentrations.

Arrhenius Equation

k = A e^{-E_a / RT}

Temperature dependence of reaction rate constants.

Gibbs Free Energy

\Delta G = \Delta H - T \Delta S

Spontaneity of chemical reactions.

Enthalpy Change

\Delta H = q_p

Heat at constant pressure.

Entropy Change

\Delta S = \frac{q_{rev}}{T}

Measure of disorder change.

Equilibrium Constant

K_c = \frac{[C]^c [D]^d}{[A]^a [B]^b}

Ratio of product to reactant concentrations.

Solubility Product

K_{sp} = [A^+]^a [B^-]^b

Equilibrium constant for sparingly soluble salts.

Half-Life

t_{1/2} = \frac{\ln 2}{\lambda}

Time for half of radioactive substance to decay.

Mathematics

Core formulas and quick explanations for mathematics.

Quadratic Formula

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Solutions of a quadratic equation.

Discriminant

D = b^2 - 4ac

Determines the nature of quadratic roots.

Arithmetic Progression Sum

S_n = \frac{n}{2}[2a + (n-1)d]

Sum of the first n terms of an AP.

Geometric Progression Sum

S_n = \frac{a(r^n - 1)}{r - 1}

Sum of the first n terms of a GP.

Pythagoras Theorem

a^2 + b^2 = c^2

Relationship between sides of a right triangle.

Area of Circle

A = \pi r^2

Area inside a circle.

Circumference of Circle

C = 2\pi r

Distance around a circle.

Volume of Sphere

V = \frac{4}{3}\pi r^3

Volume enclosed by a sphere.

Surface Area of Sphere

SA = 4\pi r^2

Surface area of a sphere.

Trigonometric Ratios

\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}} \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}} \tan\theta = \frac{\text{opposite}}{\text{adjacent}}

Basic trigonometric definitions for right triangles.

Pythagorean Identity

\sin^2\theta + \cos^2\theta = 1

Fundamental identity for sine and cosine.

Binomial Theorem

(a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k

Expansion of powers of binomials.

Logarithm Properties

\log_a (xy) = \log_a x + \log_a y \log_a \left(\frac{x}{y}\right) = \log_a x - \log_a y \log_a x^n = n \log_a x

Key rules for logarithmic functions.

Exponential Growth

A = A_0 e^{kt}

Continuous growth model.

Derivative

\frac{dy}{dx} = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

Rate of change or slope of tangent.

Integral

\int f(x) \, dx = F(x) + C

Antiderivative or area under curve.

Biology

Core formulas and quick explanations for biology.

Photosynthesis

6CO_2 + 6H_2O → C_6H_{12}O_6 + 6O_2

Plants convert carbon dioxide and water into glucose and oxygen.

Aerobic Respiration

C_6H_{12}O_6 + 6O_2 → 6CO_2 + 6H_2O + ATP

Glucose breakdown in the presence of oxygen.

Anaerobic Respiration

C_6H_{12}O_6 → 2C_3H_6O_3 + \text{energy}

Glucose breakdown without oxygen in animals.

BMI

BMI = \frac{\text{weight (kg)}}{\text{height (m)}^2}

Body mass index for weight assessment.

Hardy–Weinberg Equation

p^2 + 2pq + q^2 = 1

Allele frequency distribution in a stable population.

Cardiac Output

CO = \text{Stroke Volume} × \text{Heart Rate}

Volume of blood pumped by the heart per minute.

Respiratory Quotient

RQ = \frac{CO_2 \text{ produced}}{O_2 \text{ consumed}}

Ratio of carbon dioxide produced to oxygen consumed.

Water Potential

\Psi = \Psi_s + \Psi_p

Total water potential from solute and pressure potentials.

ATP Yield

Approx. 30–32 ATP per glucose molecule

Modern estimate of ATP produced from one glucose.

Population Growth

\frac{dN}{dt} = rN

Exponential growth in population biology.

Logistic Growth

\frac{dN}{dt} = rN \left(1 - \frac{N}{K}\right)

Growth with carrying capacity.

Carrying Capacity

K

Maximum population size an environment can sustain.

Osmotic Pressure

\Pi = CRT

Pressure difference across a semipermeable membrane.

Michaelis-Menten Equation

v = \frac{V_{max} [S]}{K_m + [S]}

Enzyme kinetics model.

Key Constants

g9.81 m/s²

G6.674 × 10^{-11} N·m^2/kg^2

c3.0 × 10^8 m/s

h6.626 × 10^{-34} J·s

N_A6.022 × 10^{23} mol^{-1}

R8.314 J·mol^{-1}·K^{-1}

k_B1.38 × 10^{-23} J/K

e1.602 × 10^{-19} C

ε_08.85 × 10^{-12} F/m

π3.14159

SI Units & Conversions Reference

A searchable, evergreen reference for SI prefixes, base and derived units, and key unit conversions. Bookmark this section for fast lookup while solving STEM problems.

Showing 20 prefixes, 24 units, and 13 conversions.

SI Prefixes

Prefix	Symbol	Factor
yotta	Y	10^{24}
zetta	Z	10^{21}
exa	E	10^{18}
peta	P	10^{15}
tera	T	10^{12}
giga	G	10^{9}
mega	M	10^{6}
kilo	k	10^{3}
hecto	h	10^{2}
deca	da	10^{1}
deci	d	10^{-1}
centi	c	10^{-2}
milli	m	10^{-3}
micro	μ	10^{-6}
nano	n	10^{-9}
pico	p	10^{-12}
femto	f	10^{-15}
atto	a	10^{-18}
zepto	z	10^{-21}
yocto	y	10^{-24}

SI Units

Symbol	Unit	Quantity	Definition
m	metre	length	SI base unit of distance
kg	kilogram	mass	SI base unit of mass
s	second	time	SI base unit of time
A	ampere	electric current	SI base unit of electric current
K	kelvin	temperature	SI base unit of thermodynamic temperature
mol	mole	amount of substance	SI base unit of amount of substance
cd	candela	luminous intensity	SI base unit of luminous intensity
N	newton	force	kg·m/s^2
Pa	pascal	pressure	N/m^2
J	joule	energy	N·m
W	watt	power	J/s
C	coulomb	electric charge	A·s
V	volt	electric potential	W/A
Ω	ohm	resistance	V/A
S	siemens	conductance	A/V
F	farad	capacitance	C/V
T	tesla	magnetic flux density	Wb/m^2
H	henry	inductance	Wb/A
lm	lumen	luminous flux	cd·sr
lx	lux	illuminance	lm/m^2
Bq	becquerel	radioactivity	1/s
Gy	gray	absorbed dose	J/kg
Sv	sievert	dose equivalent	J/kg
kat	katal	catalytic activity	mol/s

Common Conversions

From	To	Description
1 km	1000 m	Kilometre to metre
1 cm	0.01 m	Centimetre to metre
1 mm	0.001 m	Millimetre to metre
1 g	0.001 kg	Gram to kilogram
1 L	0.001 m^3	Litre to cubic metre
1 min	60 s	Minute to second
1 h	3600 s	Hour to second
1 atm	101325 Pa	Standard atmosphere to pascal
1 bar	1 × 10^5 Pa	Bar to pascal
1 eV	1.602 × 10^{-19} J	Electronvolt to joule
1 cal	4.184 J	Calorie to joule
1 hp	745.7 W	Horsepower to watt
1 °C	274.15 K	Degree Celsius to kelvin

3D Formula Explanations

Interactive 3D visualizations to help understand how formulas work. Use mouse to rotate, zoom, and pan.

Newton's Second Law: F = ma

This visualization shows a mass (blue cube) experiencing a force (red arrow), resulting in acceleration (green arrow). The force causes the mass to accelerate in the direction of the force.

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More 3D explanations coming soon for other formulas!

How to Read the Formulas

Each formula block includes the name of the concept, the equation itself, and a short explanation. Use these blocks as a quick reference while solving problems, then apply the calculators to verify your results.

This cheat sheet is designed to help you quickly recall the right formula and understand the context behind it. For best results, try to solve the problem manually first, then cross-check with the online tools on SciFi Calculators.