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
algebrax = (−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
algebrad = √((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
algebram = (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
algebray − 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
algebraaᵐ×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
algebralog(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
geometryA = πr²
Area of a circle with radius r. π ≈ 3.14159265.
Example
r=5cm: A=π×25=78.54 cm²
Circle Circumference
geometryC = 2πr = πd
Perimeter of a circle. d=diameter=2r.
Example
r=7cm: C=2π×7=43.98 cm
Triangle Area
geometryA = 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
geometryA = √(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
geometrya² + 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
geometryV = (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
geometrySA = 4πr²
Total surface area of a sphere.
Example
r=5cm: SA=4π×25=314.16 cm²
Volume of a Cylinder
geometryV = 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
geometryV = (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
geometryA = 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
financeI = 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
financeA = 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
financePV = 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
financeFV = 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
financeM = 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)
financeROI = (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
financeGPM = (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
financeYears 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
financePrice 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
statisticsx̄ = (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
statisticsσ² = 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
statisticsσ = √(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
statisticsz = (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
statisticsP(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
statisticsP(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
statisticsC(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
trigonometrysin 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
trigonometrysin²θ + 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
trigonometrya/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
trigonometryc² = 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
trigonometryA = 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
trigonometrysin(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
physicsF = 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
physicsKE = 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
physicsGPE = 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
physicsV = IR
Voltage (V) = Current (A) × Resistance (Ω). Fundamental in electronics.
Example
Current 2A, resistance 10Ω: V=2×10=20V
Speed
physicsv = d / t
Speed = distance ÷ time. Average speed over a journey.
Example
Distance 120 km in 2 hours: v=60 km/h
Density
physicsρ = m / V
Density (kg/m³) = mass (kg) ÷ volume (m³).
Example
Mass 500g, volume 250cm³: ρ=2 g/cm³
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Getting the most out of a formula reference
A formula on its own is just a string of symbols — the useful part is knowing which formula applies to your situation and what each variable represents in the real world. That's why every entry in this library pairs the formula with a plain-English description of what it does and a worked example using real numbers, rather than just listing the notation. When you're trying to identify which formula you need, it often helps to search by the shape of your problem rather than a formula's name: searching "triangle" surfaces area, Heron's formula, Pythagoras, and both the sine and cosine rules together, so you can compare which one fits the information you actually have (three sides? two angles and a side? a right angle?).
Formulas also tend to cluster into families that build on each other. The compound interest formula, present value, and future value formulas in the finance section are all variations of the same exponential growth relationship, just solved for different unknowns — once you understand one, the others follow the same logic in reverse. Similarly in trigonometry, SOH-CAH-TOA only works for right-angled triangles, which is why the sine rule and cosine rule exist to handle any triangle shape. Recognising these families makes it much easier to remember formulas long-term, because you're storing one underlying idea instead of ten disconnected strings of symbols.
If you're studying for an exam, a practical approach is to work through the examples on this page by hand before checking the answer, rather than reading the formula and example passively — recalculating each worked example yourself is a far better test of whether you actually understand a formula than recognising it when you see it. For everyday use — working out a mortgage payment, a triangle's area for a DIY project, or a discount percentage — the matching calculator linked at the bottom of this page will do the arithmetic for you once you know which formula applies.
Frequently Asked Questions
Is it better to memorise formulas or look them up?+
For exams with closed-book conditions, memorising the formulas you use often (quadratic formula, area/volume of common shapes, compound interest) pays off because recall is faster than searching. For everyday or professional use, looking a formula up once and understanding what each symbol represents is usually more valuable than rote memorisation.
Why do some formulas have more than one form?+
Many formulas are algebraically rearranged depending on which variable you need to solve for. Compound interest, for example, can be written to solve for the final amount, the principal, the rate, or the time period — they're all the same underlying relationship, just isolated for a different unknown.
What's the difference between a formula and an equation?+
An equation is any mathematical statement that two expressions are equal. A formula is a specific type of equation that expresses a general relationship between variables that holds for all valid inputs — the quadratic formula and F=ma work for any values of their variables, not just one specific case.
How accurate are the worked examples?+
Every worked example has been calculated and verified independently rather than copied from a textbook, following the accuracy standard described on our methodology page. If you spot a discrepancy, our editorial policy explains how to report it.
Put These Formulas to Work
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