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Formula collection

Areas, solids, algebra and physics: the key formulas with their symbols explained.

Areas

  • Rectangle

    where: a, b: side lengths

    A=a⋅bA = a \cdot b

  • Triangle

    where: g: base, h: matching height

    A=12⋅g⋅hA = \tfrac{1}{2} \cdot g \cdot h

  • Trapezoid

    where: a, c: parallel sides, h: height

    A=a+c2⋅hA = \tfrac{a + c}{2} \cdot h

  • Parallelogram

    where: a: side, h_a: height onto a

    A=a⋅haA = a \cdot h_a

  • Circle

    where: r: radius

    A=π⋅r2A = \pi \cdot r^2

  • Circle circumference

    where: r: radius

    U=2⋅π⋅rU = 2 \cdot \pi \cdot r
Solids

  • Cuboid

    where: a, b, c: edge lengths

    V=a⋅b⋅cV = a \cdot b \cdot c

  • Cylinder

    where: r: base radius, h: height

    V=π⋅r2⋅hV = \pi \cdot r^2 \cdot h

  • Cone

    where: r: radius, h: height

    V=13⋅π⋅r2⋅hV = \tfrac{1}{3} \cdot \pi \cdot r^2 \cdot h

  • Sphere

    where: r: radius

    V=43⋅π⋅r3V = \tfrac{4}{3} \cdot \pi \cdot r^3

  • Pyramid

    where: G: base area, h: height

    V=13⋅G⋅hV = \tfrac{1}{3} \cdot G \cdot h
Algebra

  • Pythagorean theorem

    where: a, b: legs, c: hypotenuse

    a2+b2=c2a^2 + b^2 = c^2

  • First binomial formula

    where: holds for any a, b

    (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2

  • Second binomial formula

    where: holds for any a, b

    (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2

  • Third binomial formula

    where: holds for any a, b

    (a+b)(a−b)=a2−b2(a + b)(a - b) = a^2 - b^2

  • Quadratic formula

    where: for a x^2 + b x + c = 0, a ≠ 0

    x1,2=−b±b2−4ac2ax_{1,2} = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

  • Linear function

    where: m: slope, b: y-intercept

    y=m⋅x+by = m \cdot x + b

  • Slope

    where: from two points on the line

    m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}
Percentages & interest

  • Percentage value

    where: p: rate, G: base value, W: percentage value

    W=p100⋅GW = \frac{p}{100} \cdot G

  • Interest (yearly)

    where: K: principal, p: rate, t: years

    Z=K⋅p⋅t100Z = \frac{K \cdot p \cdot t}{100}
Trigonometry

  • Sine, cosine, tangent

    where: in a right triangle, a: opposite, b: adjacent, c: hypotenuse

    sin⁡α=ac,cos⁡α=bc,tan⁡α=ab\sin\alpha = \tfrac{a}{c}, \quad \cos\alpha = \tfrac{b}{c}, \quad \tan\alpha = \tfrac{a}{b}

  • Sine rule

    where: in any triangle, side divided by sine of the opposite angle

    asin⁡α=bsin⁡β=csin⁡γ\frac{a}{\sin\alpha} = \frac{b}{\sin\beta} = \frac{c}{\sin\gamma}

  • Cosine rule

    where: in any triangle, γ: the angle between a and b

    c2=a2+b2−2abcos⁡γc^2 = a^2 + b^2 - 2ab\cos\gamma

  • Pythagorean identity

    where: holds for every angle α

    sin⁡2α+cos⁡2α=1\sin^2\alpha + \cos^2\alpha = 1
Motion & force

  • Speed

    where: s: distance, t: time

    v=stv = \frac{s}{t}

  • Acceleration

    where: Δv: change in speed, Δt: time span

    a=ΔvΔta = \frac{\Delta v}{\Delta t}

  • Newton's second law

    where: m: mass, a: acceleration

    F=m⋅aF = m \cdot a

  • Weight force

    where: g ≈ 9.81 m/s² on Earth

    FG=m⋅gF_G = m \cdot g

  • Density

    where: m: mass, V: volume

    ρ=mV\rho = \frac{m}{V}

  • Pressure

    where: F: force, A: area

    p=FAp = \frac{F}{A}
Energy & electricity

  • Kinetic energy

    where: m: mass, v: speed

    Ekin=12⋅m⋅v2E_{\text{kin}} = \tfrac{1}{2} \cdot m \cdot v^2

  • Potential energy

    where: h: height, g: gravity

    Epot=m⋅g⋅hE_{\text{pot}} = m \cdot g \cdot h

  • Work

    where: F: force along the path, s: distance

    W=F⋅sW = F \cdot s

  • Ohm's law

    where: U: voltage, R: resistance, I: current

    U=R⋅IU = R \cdot I

  • Electric power

    where: U: voltage, I: current

    P=U⋅IP = U \cdot I
Amounts & solutions

  • Amount of substance

    where: m: mass, M: molar mass

    n=mMn = \frac{m}{M}

  • Concentration

    where: n: amount, V: solution volume

    c=nVc = \frac{n}{V}

  • Ideal gas law

    where: R: gas constant, T: temperature in kelvin

    p⋅V=n⋅R⋅Tp \cdot V = n \cdot R \cdot T

  • pH value

    where: c(H⁺): hydrogen ion concentration

    pH=−log⁡10 ⁣(c(H+))\text{pH} = -\log_{10}\!\big(c(\text{H}^+)\big)
Technology & electronics

  • Resistors in series

    where: total resistance of a series circuit

    R=R1+R2R = R_1 + R_2

  • Resistors in parallel

    where: total resistance of a parallel circuit

    1R=1R1+1R2\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2}

  • Capacitor charge

    where: Q: charge, C: capacitance, U: voltage

    Q=C⋅UQ = C \cdot U

  • Gear ratio

    where: z: tooth counts, i times slower, i times more torque

    i=z2z1i = \frac{z_2}{z_1}

  • Pulley system

    where: G: weight, n: supporting rope strands

    F=GnF = \frac{G}{n}

  • Golden rule of mechanics

    where: less force means more distance, the work stays the same

    F1⋅s1=F2⋅s2F_1 \cdot s_1 = F_2 \cdot s_2