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ESAT Formulae

The 117 formulae the ESAT specification expects you to know across all five modules, grouped by topic. Each one is tagged with its spec point. Nothing outside the spec is included, even where it shows up in worked solutions.

Maths 1

M1

Units

SpeedM1.1
speed=distancetime\text{speed} = \dfrac{\text{distance}}{\text{time}}
DensityM1.1
density=massvolume\text{density} = \dfrac{\text{mass}}{\text{volume}}
PressureM1.1
pressure=forcearea\text{pressure} = \dfrac{\text{force}}{\text{area}}
M2

Number

Index law — multiplicationM2.7
am×an=am+na^m \times a^n = a^{m+n}
Index law — divisionM2.7
aman=am−n\dfrac{a^m}{a^n} = a^{m-n}
Index law — power of a powerM2.7
(am)n=amn(a^m)^n = a^{mn}
Zero indexM2.7
a0=1a^0 = 1
Negative indexM2.7
a−n=1ana^{-n} = \dfrac{1}{a^n}
Fractional indexM2.7
a1n=an,amn=(an)ma^{\frac{1}{n}} = \sqrt[n]{a}, \qquad a^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^{m}
Surd lawsM2.11
ab=a b,ab=ab\sqrt{ab} = \sqrt{a}\,\sqrt{b}, \qquad \sqrt{\dfrac{a}{b}} = \dfrac{\sqrt{a}}{\sqrt{b}}
Standard formM2.8
a×10n,1≤a<10a \times 10^{n}, \quad 1 \le a < 10
M3

Ratio and proportion

Percentage changeM3.8
% change=changeoriginal×100\%\text{ change} = \dfrac{\text{change}}{\text{original}} \times 100
Compound growth / decayM3.11
amount=P(1±r100)n\text{amount} = P\left(1 \pm \dfrac{r}{100}\right)^{n}

P start value, r % rate, n periods (+ growth, − decay)

Direct and inverse proportionM3.9
y=kxy=kxy = kx \qquad y = \dfrac{k}{x}

k constant of proportionality

M4

Algebra

Difference of two squaresM4.5
a2−b2=(a+b)(a−b)a^2 - b^2 = (a+b)(a-b)
Quadratic formulaM4.16
x=−b±b2−4ac2ax = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}

for ax² + bx + c = 0

Equation and gradient of a straight lineM4.10
y=mx+c,m=y2−y1x2−x1y = mx + c, \qquad m = \dfrac{y_2 - y_1}{x_2 - x_1}
nth term of a linear (arithmetic) sequenceM4.19
nth term=a+(n−1)dn\text{th term} = a + (n-1)d

a first term, d common difference

M5

Geometry

Formulae for the surface area and volume of spheres, cones and pyramids are provided in the exam (M5.15), so are not memorised.

Pythagoras' theoremM5.7
a2+b2=c2a^2 + b^2 = c^2

c hypotenuse

Trigonometric ratios (SOHCAHTOA)M5.18
sin⁡θ=opphyp,cos⁡θ=adjhyp,tan⁡θ=oppadj\sin\theta = \dfrac{\text{opp}}{\text{hyp}}, \quad \cos\theta = \dfrac{\text{adj}}{\text{hyp}}, \quad \tan\theta = \dfrac{\text{opp}}{\text{adj}}
Circle: circumference and areaM5.15
C=2πr=πd,A=πr2C = 2\pi r = \pi d, \qquad A = \pi r^2
Volume of a cylinderM5.15
V=πr2hV = \pi r^2 h
Area of triangle, parallelogram, trapeziumM5.14
A△=12bh,A∥=bh,Atrap=12(a+b)hA_{\triangle} = \tfrac{1}{2} b h, \quad A_{\parallel} = bh, \quad A_{\text{trap}} = \tfrac{1}{2}(a+b)h
Volume of a prismM5.14
Vprism=(cross-section area)×lengthV_{\text{prism}} = (\text{cross-section area}) \times \text{length}
Arc length and sector area (degrees)M5.16
arc=θ360×2πr,Asector=θ360×πr2\text{arc} = \dfrac{\theta}{360}\times 2\pi r, \qquad A_{\text{sector}} = \dfrac{\theta}{360}\times \pi r^2
M6

Statistics

MeanM6.3
xˉ=∑xn\bar{x} = \dfrac{\sum x}{n}
RangeM6.3
range=largest−smallest\text{range} = \text{largest} - \text{smallest}
M7

Probability

Theoretical probabilityM7.3
P(event)=favourable outcomestotal outcomesP(\text{event}) = \dfrac{\text{favourable outcomes}}{\text{total outcomes}}
ComplementM7.4
P(not A)=1−P(A)P(\text{not } A) = 1 - P(A)
OR rule (mutually exclusive)M7.7
P(A or B)=P(A)+P(B)P(A \text{ or } B) = P(A) + P(B)
AND rule (independent)M7.7
P(A and B)=P(A)×P(B)P(A \text{ and } B) = P(A) \times P(B)

Maths 2

MM1

Algebra and functions

Rationalising the denominator (conjugate)MM1.2
1a+b=a−ba2−b\dfrac{1}{a + \sqrt{b}} = \dfrac{a - \sqrt{b}}{a^2 - b}
Discriminant of a quadraticMM1.3
Δ=b2−4ac\Delta = b^2 - 4ac

Δ > 0 two real roots · Δ = 0 one repeated root · Δ < 0 no real roots

Completing the squareMM1.3
x2+bx+c=(x+b2)2−b24+cx^2 + bx + c = \left(x + \tfrac{b}{2}\right)^2 - \dfrac{b^2}{4} + c
Quadratic formulaMM1.3
x=−b±b2−4ac2ax = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}
MM2

Sequences and series

Sum of the first n natural numbersMM2.2
1+2+⋯+n=n(n+1)21 + 2 + \cdots + n = \dfrac{n(n+1)}{2}
Sum of an arithmetic seriesMM2.2
Sn=n2[ 2a+(n−1)d ]S_n = \dfrac{n}{2}\big[\,2a + (n-1)d\,\big]
Sum of a finite geometric seriesMM2.3
Sn=a(1−rn)1−rS_n = \dfrac{a(1 - r^{n})}{1 - r}
Sum to infinity of a convergent geometric seriesMM2.3
S∞=a1−r,∣r∣<1S_\infty = \dfrac{a}{1 - r}, \quad |r| < 1
Binomial expansionMM2.4
(1+x)n=1+nx+n(n−1)2!x2+⋯(1 + x)^n = 1 + nx + \dfrac{n(n-1)}{2!}x^2 + \cdots
Binomial coefficientMM2.4
(nr)=n!r! (n−r)!\binom{n}{r} = \dfrac{n!}{r!\,(n-r)!}
MM3

Coordinate geometry

Equation of a straight lineMM3.1a
y−y1=m(x−x1)y - y_1 = m(x - x_1)
General form of a straight lineMM3.1b
ax+by+c=0ax + by + c = 0
Condition for perpendicular linesMM3.1
m1m2=−1m_1 m_2 = -1

Parallel lines have equal gradients.

Circle, centre (a, b), radius rMM3.2a
(x−a)2+(y−b)2=r2(x - a)^2 + (y - b)^2 = r^2
Circle — general formMM3.2b
x2+y2+cx+dy+e=0x^2 + y^2 + cx + dy + e = 0
MM4

Trigonometry

Sine ruleMM4.1
asin⁡A=bsin⁡B=csin⁡C\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}
Cosine ruleMM4.1
a2=b2+c2−2bccos⁡Aa^2 = b^2 + c^2 - 2bc\cos A
Area of a triangleMM4.1
Area=12absin⁡C\text{Area} = \tfrac{1}{2}ab\sin C
Arc length and sector area (radians)MM4.2
s=rθ,Asector=12r2θs = r\theta, \qquad A_{\text{sector}} = \tfrac{1}{2}r^2\theta
Area of a segment (radians)MM4.2
Asegment=12r2(θ−sin⁡θ)A_{\text{segment}} = \tfrac{1}{2}r^2(\theta - \sin\theta)
Tangent identityMM4.5a
tan⁡θ=sin⁡θcos⁡θ\tan\theta = \dfrac{\sin\theta}{\cos\theta}
Pythagorean identityMM4.5b
sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1
Exact trig valuesMM4.3
θ30∘45∘60∘sin⁡121232cos⁡321212tan⁡1313\begin{array}{c|ccc} \theta & 30^\circ & 45^\circ & 60^\circ \\ \hline \sin & \frac12 & \frac{1}{\sqrt2} & \frac{\sqrt3}{2} \\ \cos & \frac{\sqrt3}{2} & \frac{1}{\sqrt2} & \frac12 \\ \tan & \frac{1}{\sqrt3} & 1 & \sqrt3 \end{array}

Also sin 0° = 0, cos 0° = 1, tan 0° = 0, sin 90° = 1, cos 90° = 0.

MM5

Exponentials and logarithms

Definition of a logarithmMM5.2a
ab=c  ⟺  b=log⁡aca^b = c \iff b = \log_a c
Log law — additionMM5.2b
log⁡ax+log⁡ay=log⁡a(xy)\log_a x + \log_a y = \log_a(xy)
Log law — subtractionMM5.2c
log⁡ax−log⁡ay=log⁡a ⁣(xy)\log_a x - \log_a y = \log_a\!\left(\dfrac{x}{y}\right)
Log law — powerMM5.2d
klog⁡ax=log⁡a ⁣(xk)k\log_a x = \log_a\!\left(x^{k}\right)
Log of a reciprocalMM5.2e
log⁡a ⁣(1x)=−log⁡ax\log_a\!\left(\dfrac{1}{x}\right) = -\log_a x
Log of the baseMM5.2f
log⁡aa=1\log_a a = 1
MM6

Differentiation

Power ruleMM6.2
ddx(xn)=n xn−1\dfrac{d}{dx}\left(x^{n}\right) = n\,x^{n-1}
Stationary pointsMM6.3
dydx=0\dfrac{dy}{dx} = 0

Minimum if d²y/dx² > 0, maximum if d²y/dx² < 0.

MM7

Integration

Power rule for integrationMM7.2
∫xn dx=xn+1n+1+c,n≠−1\int x^{n}\,dx = \dfrac{x^{n+1}}{n+1} + c, \quad n \neq -1
Fundamental Theorem of CalculusMM7.3
∫abf(x) dx=F(b)−F(a)\int_a^b f(x)\,dx = F(b) - F(a)
Trapezium ruleMM7.5
∫aby dx≈12h[(y0+yn)+2(y1+⋯+yn−1)]\int_a^b y\,dx \approx \tfrac{1}{2}h\big[(y_0 + y_n) + 2(y_1 + \cdots + y_{n-1})\big]
MM8

Graphs of functions

Graph transformationsMM8.2
y=af(x),    y=f(x)+a,    y=f(x+a),    y=f(ax)y = af(x), \;\; y = f(x)+a, \;\; y = f(x+a), \;\; y = f(ax)
Completed-square form (vertex at (−b, c))MM8.4
y=a(x+b)2+cy = a(x + b)^2 + c

Physics

P1

Electricity

CurrentP1.2
I=QtI = \dfrac{Q}{t}

I current, Q charge, t time

ResistanceP1.2
R=VIR = \dfrac{V}{I}

V voltage, I current

Voltage (energy per charge)P1.2
V=EQV = \dfrac{E}{Q}
Electrical powerP1.2
P=IV=I2RP = IV = I^2 R
Energy transferP1.2
E=VItE = VIt
Resistors in seriesP1.2
Rtotal=R1+R2+⋯R_{\text{total}} = R_1 + R_2 + \cdots
P2

Magnetism

Force on a current-carrying wireP2.3
F=BILF = BIL

B field strength, I current, L length

Transformer turns ratioP2.5
VpVs=npns\dfrac{V_p}{V_s} = \dfrac{n_p}{n_s}

p primary, s secondary

Ideal (100% efficient) transformer powerP2.5
VpIp=VsIsV_p I_p = V_s I_s
P3

Mechanics

Speed and accelerationP3.1
speed=distancetime,a=Δvt\text{speed} = \dfrac{\text{distance}}{\text{time}}, \qquad a = \dfrac{\Delta v}{t}
Equation of motionP3.1
v2−u2=2asv^2 - u^2 = 2as

This is the only kinematics (SUVAT) equation in the ESAT spec.

Hooke's lawP3.3
F=kxF = kx

k spring constant, x extension

Energy stored in a stretched springP3.3
E=12Fx=12kx2E = \tfrac{1}{2}Fx = \tfrac{1}{2}kx^2
Newton's second lawP3.4
F=maF = ma
WeightP3.5
W=mgW = mg

g ≈ 10 N kg⁻¹ on Earth

MomentumP3.6
p=mvp = mv
Force as rate of change of momentumP3.6
F=ΔpΔtF = \dfrac{\Delta p}{\Delta t}
Work doneP3.7
W=FdW = Fd
Gravitational potential energyP3.7
Ep=mghE_p = mgh
Kinetic energyP3.7
Ek=12mv2E_k = \tfrac{1}{2}mv^2
PowerP3.7
P=EtP = \dfrac{E}{t}
Percentage efficiencyP3.7
efficiency=useful outputtotal input×100%\text{efficiency} = \dfrac{\text{useful output}}{\text{total input}} \times 100\%
P4

Thermal physics

Specific heat capacityP4.4
E=mc ΔθE = mc\,\Delta\theta

c specific heat capacity, Δθ temperature change

P5

Matter

DensityP5.4
ρ=mV\rho = \dfrac{m}{V}
PressureP5.5
P=FAP = \dfrac{F}{A}
Hydrostatic pressureP5.5
P=hρgP = h\rho g

h depth, ρ density

Boyle's law (fixed temperature)P5.2
PV=constantPV = \text{constant}
Specific latent heatP5.3
E=mLE = mL

L specific latent heat

P6

Waves

Frequency and periodP6.1
f=1Tf = \dfrac{1}{T}
Wave speedP6.1
v=fλv = f\lambda

λ wavelength

Law of reflectionP6.3
θi=θr\theta_i = \theta_r
P7

Radioactivity

Mass numberP7.1
A=Z+NA = Z + N

Z protons, N neutrons

Radioactive decay by half-lifeP7.4
N=N0(12)t/t1/2N = N_0\left(\tfrac{1}{2}\right)^{t / t_{1/2}}

t₁/₂ half-life

Chemistry

Most of ESAT Chemistry is qualitative (bonding, tests, trends). The calculation formulae below — almost all from Quantitative chemistry (C4) — are the parts that are genuinely numerical.

C1

Atomic structure

Mass number = protons + neutronsC1.3
A=Z+NA = Z + N
Relative atomic massC1.6
Ar=∑(isotope mass×% abundance)100A_r = \dfrac{\sum(\text{isotope mass} \times \%\,\text{abundance})}{100}
C4

Quantitative chemistry

Relative molar massC4.1
Mr=∑ArM_r = \textstyle\sum A_r
Moles from massC4.3
n=mMrn = \dfrac{m}{M_r}

n moles, m mass (g)

Number of particles (Avogadro's number)C4.2
N=n×NA,NA=6.02×1023N = n \times N_A, \quad N_A = 6.02 \times 10^{23}
Percentage composition by massC4.4
% mass=Ar×(number of atoms)Mr×100\%\,\text{mass} = \dfrac{A_r \times (\text{number of atoms})}{M_r} \times 100
Moles of gas at rtp (V in dm³)C4.8
n=V24n = \dfrac{V}{24}

24 dm³ per mole at room temperature and pressure.

Concentration (mol dm⁻³)C4.9
c=nVc = \dfrac{n}{V}

V volume in dm³

Percentage yieldC4.11
% yield=actual yieldpredicted yield×100\%\,\text{yield} = \dfrac{\text{actual yield}}{\text{predicted yield}} \times 100
C8

Separation techniques

Retention factor (chromatography)C8.3
Rf=distance moved by substancedistance moved by solventR_f = \dfrac{\text{distance moved by substance}}{\text{distance moved by solvent}}
C9

Acids, bases and salts

A change of 1 on the pH scale corresponds to a ×10 change in H⁺ concentration (C9.1). The ESAT spec does not require pH = −log[H⁺].

C11

Energetics

Calorimetry energy changeC11.4
q=mc ΔTq = mc\,\Delta T
Bond-energy enthalpy changeC11.5
ΔH=∑Ebroken−∑Eformed\Delta H = \textstyle\sum E_{\text{broken}} - \sum E_{\text{formed}}
C13

Organic chemistry — general formulae

AlkanesC13.2
CnH2n+2C_nH_{2n+2}
AlkenesC13.3
CnH2nC_nH_{2n}
AlcoholsC13.5
CnH2n+1OHC_nH_{2n+1}OH
Carboxylic acidsC13.6
CnH2n+1COOHC_nH_{2n+1}COOH

Biology

ESAT Biology is assessed conceptually — the specification lists no formula sheet. The only genuinely quantitative skills are quadrat / belt-transect population estimates (B10.3) and genetic-cross ratios (B3–B4), which are methods rather than memorised formulae.

B10

Ecosystems

Estimating population size from quadratsB10.3
population=total areaarea sampled×(mean count per quadrat)\text{population} = \dfrac{\text{total area}}{\text{area sampled}} \times (\text{mean count per quadrat})