Formula Reference

Searchable engineering and science equations transcribed from reference sheets. Click a source icon to view the original.

Physics(13)

Average acceleration

Kinematics

a_{a v g} = \frac{Δ v}{Δ t}

Average acceleration over a time interval.

Average velocity

Kinematics

v_{a v g} = \frac{Δ d}{Δ t}

Average velocity over a time interval.

Density

Properties

ρ = \frac{m}{v}

Mass per unit volume.

Electric charge

Electromagnetism

Q = I t

Charge equals current times time.

Impulse

Mechanics

J = Δ p = \int F d t

Impulse equals change in momentum; for constant force, J = FΔt.

Kinematic equation (displacement)

Kinematics

s = v_{0} t + \frac{1}{2} a t^{2}

Displacement under constant acceleration.

Kinematic equation (velocity-displacement)

Kinematics

v^{2} = v_{0}^{2} + 2 a s

Velocity squared equals initial velocity squared plus twice acceleration times displacement.

Kinematic equation (velocity-time)

Kinematics

v = v_{0} + a t

Final velocity under constant acceleration.

Kinetic energy

Energy

T = \frac{1}{2} m v^{2}

Translational kinetic energy of a mass.

Mass-energy equivalence

Relativity

E = m c^{2}

Energy equals mass times the speed of light squared.

Newton's law of gravitation

Gravitation

F = \frac{G m _{1} m _{2}}{r ^{2}}

Gravitational force between two masses.

Newton's second law (general)

Mechanics

\sum F = \frac{d p}{d t} = ma

Net force equals the rate of change of momentum; for constant mass, F = ma.

Torque

Rotational

\sum τ = r \times F

Net torque is the cross product of position vector and force.

Chemistry(3)

Ideal Gas Law

Gases

P V = n R T

Relates pressure, volume, moles, and temperature of an ideal gas.

Molarity

Solutions

M = \frac{n}{V}

Concentration of a solute in a solution in moles per liter.

pH from hydrogen ion concentration

Acids and bases

pH = - lo g_{10} [H^{+}]

Measures acidity from the hydrogen ion concentration.

Biology(3)

Hardy-Weinberg equilibrium

Population genetics

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

Predicts allele and genotype frequencies in a non-evolving population.

Logistic population growth

Population dynamics

\frac{d N}{d t} = r N (1 - \frac{N}{K})

Models population growth limited by carrying capacity.

Surface-area-to-volume ratio (cube)

Cell biology

\frac{S A}{V} = \frac{6}{s}

Ratio that constrains cell size and nutrient exchange.

Geology(3)

Bulk density

Physical properties

ρ = \frac{m}{V}

Mass per unit volume of a rock or soil sample.

Darcy's law

Groundwater flow

Q = - k A \frac{d h}{d l}

Flow rate through a porous medium driven by hydraulic gradient.

Porosity

Rock properties

n = \frac{V _{v}}{V _{t}}

Fraction of void volume to total volume.

Algebra(3)

Distance between two points

Coordinate geometry

d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}

Euclidean distance between two points in the plane.

Quadratic formula

Polynomials

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

Solutions to a quadratic equation ax² + bx + c = 0.

Slope-intercept form

Linear equations

y = m x + b

Equation of a straight line in slope-intercept form.

Geometry & Trigonometry(38)

Arc length

Circles

s = r θ

Arc length for a central angle in radians.

Area of a circle

Circles

A = π r^{2}

Area enclosed by a circle of radius r.

Area of a circular sector

Circles

A = \frac{1}{2} r^{2} θ

Area swept by a central angle in radians.

Area of a parallelogram

Polygons

A = bh

Area of a parallelogram from base and perpendicular height.

Area of a regular polygon

Polygons

A = \frac{1}{2} P a

Area from perimeter and apothem of a regular polygon.

Area of a trapezoid

Polygons

A = \frac{1}{2} (b_{1} + b_{2}) h

Area of a trapezoid from its two parallel bases and height.

Area of a triangle

Triangles

A = \frac{1}{2} bh

Area of a triangle from base and height.

Circumference of a circle

Circles

C = 2 π r

Perimeter of a circle of radius r.

Cosine double-angle identity

Trigonometric identities

cos (2 θ) = cos^{2} θ - sin^{2} θ

Double-angle identity for cosine.

Equation of a circle

Circles

(x - h)^{2} + (y - k)^{2} = r^{2}

Standard equation of a circle centered at (h,k) with radius r.

General form of a line

Coordinate geometry

A x + B y + C = 0

General linear equation in two variables.

Heron's formula

Triangles

A = s (s - a) (s - b) (s - c), s = \frac{a + b + c}{2}

Area of a triangle from its three side lengths.

Interior angle of a regular polygon

Polygons

θ = \frac{( n - 2 ) \times 18 0 ^{\circ}}{n}

Each interior angle of a regular n-gon.

Law of cosines

Triangles

c^{2} = a^{2} + b^{2} - 2 ab cos C

Generalizes the Pythagorean theorem to non-right triangles.

Law of sines

Triangles

\frac{a}{sin A} = \frac{b}{sin B} = \frac{c}{sin C}

Relates side lengths to the sines of opposite angles in any triangle.

Midpoint formula

Coordinate geometry

M = (\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})

Coordinates of the midpoint of a segment.

Point-slope form of a line

Coordinate geometry

y - y_{1} = m (x - x_{1})

Equation of a line through a point with given slope.

Pythagorean theorem

Triangles

a^{2} + b^{2} = c^{2}

Relates the sides of a right triangle.

Pythagorean trigonometric identity

Trigonometric identities

sin^{2} θ + cos^{2} θ = 1

Fundamental identity relating sine and cosine.

Sine double-angle identity

Trigonometric identities

sin (2 θ) = 2 sin θ cos θ

Double-angle identity for sine.

Sine of a sum

Trigonometric identities

sin (A + B) = sin A cos B + cos A sin B

Sum identity for sine.

Slope of a line

Coordinate geometry

m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}

Slope between two points in the plane.

Sum of interior angles of a polygon

Polygons

S = (n - 2) \times 18 0^{\circ}

Total interior angle measure of an n-sided polygon.

Surface area of a cone

3D shapes

S A = π r^{2} + π r l

Total surface area (base + lateral) of a right circular cone.

Surface area of a cube

3D shapes

S A = 6 s^{2}

Total surface area of a cube.

Surface area of a cylinder

3D shapes

S A = 2 π r^{2} + 2 π r h

Total surface area (bases + lateral) of a right circular cylinder.

Surface area of a rectangular prism

3D shapes

S A = 2 (l w + l h + w h)

Total surface area of a rectangular box.

Surface area of a sphere

3D shapes

S A = 4 π r^{2}

Total surface area of a sphere.

Tangent in terms of sine and cosine

Trigonometric identities

tan θ = \frac{sin θ}{cos θ}

Definition of tangent as the ratio of sine to cosine.

Triangle angle sum

Triangles

A + B + C = 18 0^{\circ}

Interior angles of any triangle sum to 180 degrees.

Triangle area (two sides, included angle)

Triangles

A = \frac{1}{2} ab sin C

Area from two sides and the sine of the included angle.

Volume of a cone

3D shapes

V = \frac{1}{3} π r^{2} h

Volume of a right circular cone.

Volume of a cube

3D shapes

V = s^{3}

Volume of a cube from its edge length.

Volume of a cylinder

3D shapes

V = π r^{2} h

Volume of a right circular cylinder.

Volume of a pyramid

3D shapes

V = \frac{1}{3} π r^{2} h

Volume of a pyramid from base area and height (general form V = (1/3)Bh).

Volume of a rectangular prism

3D shapes

V = l w h

Volume of a rectangular box from length, width, and height.

Volume of a sphere

3D shapes

V = \frac{4}{3} π r^{3}

Volume enclosed by a sphere of radius r.

Volume of a spherical shell

3D shapes

V = \frac{4}{3} π R^{3} - \frac{4}{3} π r^{3}

Volume between two concentric spheres of radii R and r.

Calculus & Diff. Eqns(3)

Chain rule

Differentiation

\frac{d y}{d x} = \frac{d y}{d u} \cdot \frac{d u}{d x}

Derivative of a composite function.

Fundamental theorem of calculus

Integration

\int_{a}^{b} f (x) d x = F (b) - F (a)

Links definite integration to antiderivatives.

Power rule (derivative)

Differentiation

\frac{d}{d x} [x^{n}] = n x^{n - 1}

Derivative of a power function.

Statistics(3)

Sample mean

Descriptive statistics

\overset{x}{ˉ} = \frac{\sum x _{i}}{n}

Arithmetic average of a set of values.

Standard deviation

Dispersion

σ = \frac{\sum ( x _{i} - μ ) ^{2}}{N}

Spread of a population about its mean.

Z-score

Normal distribution

z = \frac{x - μ}{σ}

Number of standard deviations a value is from the mean.

Statics(3)

Force equilibrium

Equilibrium

\sum F = 0

Sum of forces on a body in static equilibrium is zero.

Moment equilibrium

Equilibrium

\sum M = 0

Sum of moments on a body in static equilibrium is zero.

Moment of a force

Moments

M = F \cdot d

Moment produced by a force about a point.

Dynamics(11)

Circular acceleration (angular)

Kinematics

α = \frac{d ω}{d t}

Angular acceleration is the time derivative of angular velocity.

Circular velocity (angular)

Kinematics

ω = \frac{d θ}{d t}

Angular velocity is the time derivative of angular position.

Kinetic energy (rectilinear)

Work-Energy

E = \frac{1}{2} m v^{2}

Translational kinetic energy.

Kinetic energy (rotational)

Work-Energy

E = \frac{1}{2} I ω^{2}

Rotational kinetic energy.

Newton's second law (rectilinear)

Kinetics

F = ma

Force equals mass times acceleration.

Newton's second law (rotational)

Kinetics

τ = I α

Torque equals moment of inertia times angular acceleration.

Rectilinear acceleration

Kinematics

a = \frac{d v}{d t}

Acceleration is the time derivative of velocity.

Rectilinear position

Kinematics

x

Position variable for rectilinear motion.

Rectilinear velocity

Kinematics

v = \frac{d x}{d t}

Velocity is the time derivative of position.

Work (rectilinear)

Work-Energy

W = \int F d x

Work is the integral of force over displacement.

Work (rotational)

Work-Energy

W = \int τ d θ

Rotational work is the integral of torque over angular displacement.

Mechanics of Materials(2)

Screw ideal mechanical advantage

Simple machines

I M A = \frac{C}{Pitch}

Ideal mechanical advantage of a screw equals circumference divided by pitch.

Screw pitch from TPI

Simple machines

Pitch = \frac{1}{TPI}

Pitch is the reciprocal of threads per inch.

Fluid Mechanics(2)

Continuity equation (mass conservation)

Continuity

\frac{d m}{d t} = \overset{m}{˙}_{in} - \overset{m}{˙}_{o u t}

Rate of mass storage equals mass flow rate in minus mass flow rate out.

Mass flow rate

Continuity

\overset{m}{˙} = ρ A v

Mass flow rate through a cross-section: density × area × velocity.

Geotechnical Engineering(17)

Bulk unit weight (general)

Phase relations

r_{b} = \frac{( G + s \cdot e ) \cdot r _{w}}{1 + e}

Bulk unit weight accounting for degree of saturation.

Degree of saturation

Basic properties

S = \frac{V _{w}}{V _{v}} \times 100%

Percentage of voids filled with water. Typical range: 10–100%.

Degree of saturation × specific gravity

Phase relations

s . e = w . G

Relates degree of saturation, void ratio, water content, and specific gravity.

Density

Basic properties

ρ = \frac{M}{V}

Mass density: total mass per total volume. Typical range: 1.4–2.2 g/cm³.

Dry density

Basic properties

ρ_{d} = \frac{M _{s}}{V}

Dry density: mass of solids per total volume. Typical range: 1.2–2.0 g/cm³.

Dry unit weight

Basic properties

γ_{d} = \frac{W _{s}}{V}

Weight of solids per total volume. Typical range: 12–20 kN/m³.

Dry unit weight from bulk

Phase relations

r_{d} = \frac{r _{b}}{1 + w}

Dry unit weight derived from bulk unit weight and water content.

Dry unit weight from specific gravity

Phase relations

r_{d} = \frac{G \cdot r _{w}}{1 + e}

Dry unit weight expressed in terms of specific gravity, unit weight of water, and void ratio.

Porosity

Basic properties

n = \frac{V _{v}}{V} \times 100%

Ratio of void volume to total volume, as a percentage. Typical range: 25–60%.

Porosity from void ratio

Phase relations

n = \frac{e}{1 + e}

Converts void ratio e to porosity n.

Saturated unit weight

Phase relations

r_{s a t} = \frac{( G + e ) \cdot r _{w}}{1 + e}

Unit weight when soil is fully saturated (s = 1).

Specific gravity

Basic properties

G_{s} = \frac{ρ _{s}}{ρ _{w}} = \frac{W _{s}}{γ _{w} \cdot V _{s}}

Ratio of solids density to water density. Typical range: 2.65–2.80.

Submerged (buoyant) unit weight

Phase relations

r^{'} = \frac{( G - 1 ) \cdot r _{w}}{1 + e}

Effective unit weight of soil below the water table.

Unit weight

Basic properties

γ = \frac{W}{V}

Weight per total volume. Typical range: 14–22 kN/m³.

Void ratio

Basic properties

e = \frac{V _{v}}{V _{s}}

Ratio of void volume to solid volume. Typical range: 0.4–1.5.

Void ratio from porosity

Phase relations

e = \frac{n}{1 - n}

Converts porosity n to void ratio e. Valid for 0 < n < 1.

Water content

Basic properties

w = \frac{M _{w}}{M _{s}} \times 100%

Ratio of water mass to solids mass, expressed as a percentage. Typical range: 10–50%.

Structural Engineering(6)

Simply supported beam — central point load (max deflection)

Beam formulas

Δ_{ma x} = \frac{P L ^{3}}{48 E I}

Maximum deflection at midspan for a central point load.

Simply supported beam — central point load (max moment)

Beam formulas

M_{ma x} = \frac{P L}{4}

Maximum bending moment at the point of load for a central point load.

Simply supported beam — central point load (reactions)

Beam formulas

R_{A} = R_{B} = \frac{P}{2}

Reactions for a simply supported beam with a central point load P.

Simply supported beam — UDL (max deflection)

Beam formulas

Δ_{ma x} = \frac{5 w L ^{4}}{384 E I}

Maximum deflection at center for a UDL.

Simply supported beam — UDL (max moment)

Beam formulas

M_{ma x} = \frac{w L ^{2}}{8}

Maximum bending moment at center for a UDL.

Simply supported beam — UDL (reactions)

Beam formulas

R_{A} = R_{B} = \frac{w L}{2}

Reactions for a simply supported beam under uniformly distributed load w.

Environmental & Water Resources(3)

Continuity equation

Fluid flow

Q = A V

Conservation of mass for incompressible flow.

Manning's equation

Open channel flow

Q = \frac{1}{n} A R^{2/3} S^{1/2}

Flow rate in an open channel from geometry and roughness.

Rational method (peak runoff)

Hydrology

Q = C i A

Peak runoff from a small drainage area.

Construction Management & BIM(3)

Cost Performance Index

Earned value management

C P I = \frac{E V}{A C}

Cost efficiency of work performed on a project.

Schedule Performance Index

Earned value management

S P I = \frac{E V}{P V}

Schedule efficiency of work performed on a project.

Total float

Scheduling

T F = L S - E S

Slack time available for a scheduled activity.

Mechanical Engineering(3)

Bernoulli's equation

Fluid mechanics

\frac{P}{ρ} + \frac{V ^{2}}{2} + g z = const

Energy conservation along a streamline for steady, incompressible, frictionless flow.

Power

Energy

P = \frac{W}{t}

Rate of doing work.

Thermal efficiency

Thermodynamics

η = \frac{W _{n e t}}{Q _{in}}

Net work output divided by heat input.

Electrical Engineering(3)

Capacitor charge

Capacitance

Q = C V

Charge stored on a capacitor.

Electrical power

Power

P = V I

Power dissipated by a circuit element.

Ohm's law

Circuits

V = I R

Voltage across a resistor equals current times resistance.

Chemical Engineering(3)

Fick's first law

Mass transfer

J = - D \frac{d C}{d x}

Diffusion flux driven by a concentration gradient.

Heat transfer rate

Heat transfer

Q = U A Δ T

Overall heat transfer across an area for a temperature difference.

Reynolds number

Transport phenomena

R e = \frac{ρ v D}{μ}

Ratio of inertial to viscous forces, indicating flow regime.

Computer & Information Engineering(3)

Amdahl's law

Parallel computing

S = \frac{1}{( 1 - p ) + \frac{p}{n}}

Maximum speedup from parallelizing a fraction p of a task.

Clock frequency

Digital systems

f = \frac{1}{T}

Frequency of a periodic signal from its period.

Shannon entropy

Information theory

H = - \sum p (x) lo g_{2} p (x)

Average information content of a random variable.

AI, Quantum & Data Science(3)

Gradient descent update

Optimization

θ = θ - α \nabla J (θ)

Iterative parameter update along the negative gradient of the loss.

Mean squared error

Loss functions

M S E = \frac{1}{n} \sum (y - \overset{y}{^})^{2}

Average squared difference between predictions and true values.

Sigmoid activation

Neural networks

σ (x) = \frac{1}{1 + e ^{- x}}

Squashes a real input into the range (0, 1).