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Maths Formula Library

A complete, searchable reference covering over 50 essential formulas across algebra, geometry, finance, statistics, trigonometry, and physics. Every formula includes a plain-English explanation and a worked example so you can see it in action.

How to Use This Library

Use the search box to find any formula by name, keyword, or symbol — for example, try "quadratic", "compound interest", or "pythagorean". Use the category buttons to browse by subject. Each formula card shows the formula itself, a description of what it does and when to use it, and a concrete worked example with real numbers.

This library is designed for students revising for exams, professionals who need a quick reference, and anyone who wants to understand the maths behind a calculation rather than just get a result. If you want to apply any of these formulas immediately, use the links at the bottom of the page to open the relevant calculator.

Finance Formulas

Compound interest, loan amortisation, VAT, percentage change, profit margin, and more — the formulas behind every financial decision.

Geometry & Algebra

Area, perimeter, volume of 3D shapes, the quadratic formula, slope, logarithms, and the binomial theorem — core school and university maths.

Statistics & Trigonometry

Mean, median, standard deviation, z-scores, sin/cos/tan rules, and the unit circle — the foundation of data science and applied maths.

Maths Formula Library

A complete, searchable reference of mathematical formulas — from basic algebra to finance and physics. Each formula includes a clear explanation and worked example.

formulas found

Quadratic Formula

algebra
x = (−b ± √(b²−4ac)) / 2a

Solves ax²+bx+c=0. Discriminant b²−4ac: positive→2 real roots, zero→1 root, negative→no real roots.

Example

x²−5x+6=0: a=1,b=−5,c=6 → x=(5±√1)/2 → x=3 or x=2

Distance Formula

algebra
d = √((x₂−x₁)² + (y₂−y₁)²)

Distance between two points (x₁,y₁) and (x₂,y₂) in a 2D plane.

Example

Points (1,2) and (4,6): d=√(9+16)=√25=5

Slope of a Line

algebra
m = (y₂−y₁) / (x₂−x₁)

Gradient (steepness) of a line through two points. Positive=upward, Negative=downward.

Example

Through (0,1) and (3,7): m=(7−1)/(3−0)=2

Point-Slope Form

algebra
y − y₁ = m(x − x₁)

Equation of a line with slope m through point (x₁,y₁).

Example

Slope 2 through (1,3): y−3=2(x−1) → y=2x+1

Laws of Exponents

algebra
aᵐ×aⁿ=aᵐ⁺ⁿ · aᵐ÷aⁿ=aᵐ⁻ⁿ · (aᵐ)ⁿ=aᵐⁿ · a⁰=1

Core rules for manipulating powers and exponents.

Example

2³×2²=2⁵=32 · (3²)³=3⁶=729

Logarithm Rules

algebra
log(ab)=log(a)+log(b) · log(a/b)=log(a)−log(b) · log(aⁿ)=n×log(a)

Rules for manipulating logarithms. Change of base: logₐ(b)=log(b)/log(a).

Example

log(100)=log(10×10)=log(10)+log(10)=1+1=2

Binomial Theorem

algebra
(a+b)ⁿ = Σ C(n,k) × aⁿ⁻ᵏ × bᵏ

Expands (a+b)ⁿ. C(n,k)=n!/(k!(n-k)!) is the binomial coefficient.

Example

(a+b)²=a²+2ab+b² · (a+b)³=a³+3a²b+3ab²+b³

Circle Area

geometry
A = πr²

Area of a circle with radius r. π ≈ 3.14159265.

Example

r=5cm: A=π×25=78.54 cm²

Circle Circumference

geometry
C = 2πr = πd

Perimeter of a circle. d=diameter=2r.

Example

r=7cm: C=2π×7=43.98 cm

Triangle Area

geometry
A = ½ × base × height

Area of any triangle. Height must be perpendicular to the base.

Example

base=8, height=5: A=½×8×5=20

Heron's Formula

geometry
A = √(s(s−a)(s−b)(s−c)) where s=(a+b+c)/2

Triangle area from side lengths a, b, c. s is the semi-perimeter.

Example

Sides 3,4,5: s=6 → A=√(6×3×2×1)=√36=6

Pythagoras' Theorem

geometry
a² + b² = c²

For right-angled triangles. c is the hypotenuse (longest side).

Example

a=3, b=4: c=√(9+16)=√25=5

Volume of a Sphere

geometry
V = (4/3)πr³

Volume enclosed by a sphere of radius r.

Example

r=3cm: V=(4/3)π×27=113.1 cm³

Surface Area of a Sphere

geometry
SA = 4πr²

Total surface area of a sphere.

Example

r=5cm: SA=4π×25=314.16 cm²

Volume of a Cylinder

geometry
V = πr²h

Volume of a cylinder with radius r and height h.

Example

r=3, h=10: V=π×9×10=282.7

Volume of a Cone

geometry
V = (1/3)πr²h

Volume of a cone with base radius r and height h.

Example

r=4, h=9: V=(1/3)π×16×9=150.8

Trapezoid Area

geometry
A = ½(a+b)×h

Area of a trapezoid with parallel sides a and b, height h.

Example

a=6, b=10, h=4: A=½×16×4=32

Simple Interest

finance
I = P × r × t

Interest on the principal only. P=principal, r=rate (decimal), t=time (years).

Example

£5,000 at 4% for 3 years: I=5000×0.04×3=£600

Compound Interest

finance
A = P(1 + r/n)^(nt)

A=final amount, P=principal, r=annual rate, n=compounds/year, t=years.

Example

£1,000 at 5% monthly for 10 years: A=1000×(1.004167)¹²⁰=£1,647

Present Value

finance
PV = FV / (1 + r)ⁿ

Value today of a future amount FV, discounted at rate r over n periods.

Example

£10,000 in 5 years at 5%: PV=10000/(1.05)⁵=£7,835

Future Value

finance
FV = PV × (1 + r)ⁿ

Value of a current amount PV in the future, growing at rate r over n periods.

Example

£5,000 at 7% for 10 years: FV=5000×(1.07)¹⁰=£9,836

Mortgage Payment

finance
M = P × [r(1+r)ⁿ] / [(1+r)ⁿ−1]

Monthly payment M on a loan P at monthly rate r over n months.

Example

£200,000 at 4% for 25yr: r=0.00333, n=300 → M≈£1,055/mo

Return on Investment (ROI)

finance
ROI = (Gain − Cost) / Cost × 100%

Percentage return on an investment. Higher is better.

Example

Bought for £800, sold for £1,200: ROI=(400/800)×100=50%

Gross Profit Margin

finance
GPM = (Revenue − COGS) / Revenue × 100%

Percentage of revenue retained after direct costs. COGS=cost of goods sold.

Example

Revenue £100k, COGS £60k: GPM=40/100×100=40%

Rule of 72

finance
Years to double ≈ 72 / annual rate (%)

Quick mental estimate: how long for an investment to double at a given annual rate.

Example

At 6% annual return: 72÷6=12 years to double

Percentage Change

finance
% change = (new − old) / old × 100

How much a value has changed relative to its starting point.

Example

Price goes from £80 to £100: (100−80)/80×100=+25%

VAT Calculation

finance
Price inc. VAT = net × (1 + rate) · Net = gross / (1 + rate)

Add or remove VAT. UK standard rate is 20%.

Example

Net £50, 20% VAT: gross=50×1.2=£60 · Remove: 72/1.2=£60

Arithmetic Mean

statistics
x̄ = (x₁ + x₂ + ... + xₙ) / n

Average value of a data set. Sum of all values divided by number of values.

Example

Dataset {2,4,6,8,10}: mean=(2+4+6+8+10)/5=30/5=6

Variance

statistics
σ² = Σ(xᵢ − x̄)² / n

Measures how spread out data is from the mean. Use n−1 for sample variance.

Example

Data {2,4,6}: mean=4, variance=((4+0+4)/3)=2.67

Standard Deviation

statistics
σ = √(Σ(xᵢ − x̄)² / n)

Spread of data around the mean. Square root of variance. Common in statistics and science.

Example

Variance=4: SD=√4=2

Z-Score

statistics
z = (x − μ) / σ

How many standard deviations x is from the mean μ. Used to compare data across distributions.

Example

Score 75, mean 70, SD 5: z=(75−70)/5=1.0 (1 SD above mean)

Probability

statistics
P(A) = favourable outcomes / total outcomes

Basic probability. P(A)+P(A′)=1. Independent events: P(A and B)=P(A)×P(B).

Example

Roll a die, P(even)=3/6=0.5=50%

Permutations

statistics
P(n,r) = n! / (n−r)!

Number of ordered arrangements of r items from n. Order matters.

Example

P(5,2)=5!/3!=20 ordered pairs from 5 items

Combinations

statistics
C(n,r) = n! / (r!(n−r)!)

Number of unordered selections of r items from n. Order does not matter.

Example

C(5,2)=5!/(2!3!)=10 pairs from 5 items

SOH-CAH-TOA

trigonometry
sin θ=Opp/Hyp · cos θ=Adj/Hyp · tan θ=Opp/Adj

Core trigonometric ratios for right-angled triangles.

Example

Right triangle, angle 30°, hyp=10: opposite=10×sin(30°)=5

Pythagorean Identity

trigonometry
sin²θ + cos²θ = 1

Fundamental identity derived from Pythagoras. Also: 1+tan²θ=sec²θ

Example

If sinθ=0.6, then cosθ=√(1−0.36)=√0.64=0.8

Sine Rule

trigonometry
a/sin A = b/sin B = c/sin C

Relates sides and angles of any triangle. Use when two angles and a side are known.

Example

A=30°, a=5, B=60°: b=5×sin60°/sin30°=5√3≈8.66

Cosine Rule

trigonometry
c² = a² + b² − 2ab×cos(C)

Generalised Pythagoras. Use when three sides, or two sides and included angle, are known.

Example

a=5, b=7, C=60°: c²=25+49−35=39 → c=6.24

Area via Sine

trigonometry
A = ½ab×sin(C)

Area of a triangle when two sides (a,b) and included angle C are known.

Example

a=8, b=5, C=60°: A=½×8×5×sin60°=17.32

Double Angle Formulas

trigonometry
sin(2θ)=2sinθcosθ · cos(2θ)=cos²θ−sin²θ · tan(2θ)=2tanθ/(1−tan²θ)

Express trig functions of double angles in terms of single angles.

Example

sin(60°)=2×sin30°×cos30°=2×0.5×0.866=0.866=√3/2

Newton's Second Law

physics
F = ma

Force (N) = mass (kg) × acceleration (m/s²). Foundation of classical mechanics.

Example

10 kg object, 3 m/s² acceleration: F=10×3=30 N

Kinetic Energy

physics
KE = ½mv²

Energy of a moving object. m=mass (kg), v=velocity (m/s). Result in Joules.

Example

2 kg at 10 m/s: KE=½×2×100=100 J

Gravitational Potential Energy

physics
GPE = mgh

Energy due to height. m=mass, g=9.81 m/s², h=height in metres.

Example

5 kg raised 3 m: GPE=5×9.81×3=147.15 J

Ohm's Law

physics
V = IR

Voltage (V) = Current (A) × Resistance (Ω). Fundamental in electronics.

Example

Current 2A, resistance 10Ω: V=2×10=20V

Speed

physics
v = d / t

Speed = distance ÷ time. Average speed over a journey.

Example

Distance 120 km in 2 hours: v=60 km/h

Density

physics
ρ = m / V

Density (kg/m³) = mass (kg) ÷ volume (m³).

Example

Mass 500g, volume 250cm³: ρ=2 g/cm³

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