Complete Formula Reference
All Math Formulas A to Z
Every formula you need in one place: percentage, algebra, geometry, trigonometry, logarithms, statistics, calculus basics, finance and everyday science. Each formula comes with a quick worked example, so you can see exactly how the numbers go in.
Want to check an answer? Type the expression into the free Casio scientific calculator online, convert percentages with the free online percentage calculator, or learn the details in the percentage formula guide.
Notation: r = radius, h = height, b = base, l = length, w = width, s = side, n = count, P = principal amount, and π ≈ 3.14159. Results are rounded to two decimals.
Jump to a letter, then pick a formula.
Arithmetic & Percentage
| Formula name | Formula | Worked example |
|---|
| Percentage | p = (a ÷ b) × 100 | (45 ÷ 60) × 100 = 75% |
| Percentage of a number | x = (p ÷ 100) × N | (20 ÷ 100) × 150 = 30 |
| Percentage change | ((v2 − v1) ÷ v1) × 100 | ((120 − 100) ÷ 100) × 100 = 20% |
| Discount percentage | ((P0 − P1) ÷ P0) × 100 | ((2500 − 1875) ÷ 2500) × 100 = 25% |
| Profit percentage | ((S − C) ÷ C) × 100 | ((125 − 100) ÷ 100) × 100 = 25% |
| Least common multiple (LCM) | LCM(a, b) = (a × b) ÷ GCD(a, b) | LCM(12, 18) = (12 × 18) ÷ 6 = 36 |
Algebra
| Formula name | Formula | Worked example |
|---|
| Quadratic formula | x = (−b ± √(b2 − 4ac)) ÷ 2a | x2 − 5x + 6 = 0 → x = 2 or x = 3 |
| Discriminant | D = b2 − 4ac | D = (−5)2 − 4(1)(6) = 1 |
| Square of a sum | (a + b)2 = a2 + 2ab + b2 | (3 + 4)2 = 9 + 24 + 16 = 49 |
| Square of a difference | (a − b)2 = a2 − 2ab + b2 | (7 − 2)2 = 49 − 28 + 4 = 25 |
| Difference of squares | a2 − b2 = (a + b)(a − b) | 102 − 62 = 16 × 4 = 64 |
| Cube of a sum | (a + b)3 = a3 + 3a2b + 3ab2 + b3 | (2 + 1)3 = 8 + 12 + 6 + 1 = 27 |
| n-th term of an arithmetic sequence | an = a1 + (n − 1)d | a1 = 3, d = 4, n = 10 → a10 = 39 |
| Sum of an arithmetic sequence | Sn = (n ÷ 2)(a1 + an) | 1 + 2 + … + 10 = (10 ÷ 2)(1 + 10) = 55 |
| n-th term of a geometric sequence | an = a1 × rn−1 | a1 = 2, r = 3, n = 5 → a5 = 162 |
| Sum of a geometric sequence | Sn = a1(1 − rn) ÷ (1 − r) | a1 = 2, r = 3, n = 4 → S4 = 80 |
| Slope of a line | m = (y2 − y1) ÷ (x2 − x1) | (1, 2), (4, 8) → m = 6 ÷ 3 = 2 |
| Equation of a line | y = mx + c | m = 2, c = 1, x = 3 → y = 7 |
| Distance between two points | d = √((x2 − x1)2 + (y2 − y1)2) | (0, 0), (3, 4) → d = 5 |
| Midpoint | M = ((x1 + x2) ÷ 2, (y1 + y2) ÷ 2) | (2, 4), (6, 10) → M = (4, 7) |
Geometry: Shapes & Areas
| Formula name | Formula | Worked example |
|---|
| Area of a square | A = s2 | s = 6 → A = 36 |
| Area of a rectangle | A = l × w | 8 × 5 = 40 |
| Perimeter of a rectangle | P = 2(l + w) | 2(8 + 5) = 26 |
| Area of a triangle | A = ½ × b × h | ½ × 10 × 6 = 30 |
| Heron's formula | A = √(s(s − a)(s − b)(s − c)), s = (a + b + c) ÷ 2 | 3, 4, 5 → s = 6, A = √(6 × 3 × 2 × 1) = 6 |
| Area of a circle | A = πr2 | r = 5 → A ≈ 78.54 |
| Circumference of a circle | C = 2πr | r = 5 → C ≈ 31.42 |
| Area of a trapezoid | A = ½(a + b)h | ½(6 + 10) × 4 = 32 |
| Area of a parallelogram | A = b × h | 7 × 3 = 21 |
| Area of a rhombus | A = ½ × d1 × d2 | ½ × 8 × 6 = 24 |
| Pythagorean theorem | a2 + b2 = c2 | 32 + 42 = 52 |
| Interior angles of a polygon | S = (n − 2) × 180° | n = 6 → S = 720° |
| Arc length | L = (θ ÷ 360) × 2πr | θ = 90°, r = 4 → L ≈ 6.28 |
Geometry: Volume & Surface Area
| Formula name | Formula | Worked example |
|---|
| Volume of a cube | V = s3 | s = 3 → V = 27 |
| Volume of a cuboid | V = l × w × h | 4 × 3 × 2 = 24 |
| Volume of a cylinder | V = πr2h | r = 3, h = 5 → V ≈ 141.37 |
| Volume of a cone | V = ⅓πr2h | r = 3, h = 5 → V ≈ 47.12 |
| Volume of a sphere | V = (4 ÷ 3)πr3 | r = 3 → V ≈ 113.10 |
| Surface area of a sphere | A = 4πr2 | r = 3 → A ≈ 113.10 |
| Surface area of a cylinder | A = 2πr(r + h) | r = 3, h = 5 → A ≈ 150.80 |
Trigonometry
| Formula name | Formula | Worked example |
|---|
| Pythagorean identity | sin2θ + cos2θ = 1 | θ = 30°: 0.25 + 0.75 = 1 |
| Tangent identity | tan θ = sin θ ÷ cos θ | θ = 45° → tan θ = 1 |
| SOH-CAH-TOA (right triangle) | sin θ = O ÷ H, cos θ = A ÷ H, tan θ = O ÷ A | O = 3, A = 4, H = 5 → 0.6, 0.8, 0.75 |
| Law of sines | a ÷ sin A = b ÷ sin B = c ÷ sin C | a = 5, A = 30° → 5 ÷ sin 30° = 10 |
| Law of cosines | c2 = a2 + b2 − 2ab cos C | a = 3, b = 4, C = 90° → c2 = 25 |
| Degrees to radians | rad = deg × π ÷ 180 | 90° × π ÷ 180 ≈ 1.5708 |
| Double-angle sine | sin 2θ = 2 sin θ cos θ | θ = 30° → 2 × 0.5 × 0.866 ≈ 0.866 |
| Double-angle cosine | cos 2θ = cos2θ − sin2θ | θ = 30° → 0.75 − 0.25 = 0.5 |
Exponents & Logarithms
| Formula name | Formula | Worked example |
|---|
| Product of powers | am × an = am+n | 23 × 24 = 27 = 128 |
| Quotient of powers | am ÷ an = am−n | 56 ÷ 52 = 54 = 625 |
| Power of a power | (am)n = amn | (23)2 = 26 = 64 |
| Negative exponent | a−n = 1 ÷ an | 2−3 = 1 ÷ 8 = 0.125 |
| Logarithm of a product | log(xy) = log x + log y | log 100 = log 10 + log 10 = 2 |
| Logarithm of a quotient | log(x ÷ y) = log x − log y | log(1000 ÷ 10) = 3 − 1 = 2 |
| Logarithm of a power | log(xn) = n log x | log(103) = 3 × 1 = 3 |
| Change of base | loga b = log b ÷ log a | log2 8 = log 8 ÷ log 2 = 3 |
Statistics & Probability
| Formula name | Formula | Worked example |
|---|
| Mean (average) | μ = Σx ÷ n | 2, 4, 6, 8 → μ = 20 ÷ 4 = 5 |
| Variance | σ2 = Σ(x − μ)2 ÷ n | 2, 4, 6, 8 → σ2 = 20 ÷ 4 = 5 |
| Standard deviation | σ = √(σ2) | σ = √5 ≈ 2.236 |
| Probability | P(E) = n(E) ÷ n(S) | P(3) = 1 ÷ 6 ≈ 0.167 |
| Factorial | n! = n × (n − 1) × … × 1 | 5! = 120 |
| Permutation | P(n, r) = n! ÷ (n − r)! | P(5, 2) = 120 ÷ 6 = 20 |
| Combination | C(n, r) = n! ÷ (r!(n − r)!) | C(5, 2) = 120 ÷ (2 × 6) = 10 |
| Z-score | z = (x − μ) ÷ σ | x = 85, μ = 70, σ = 5 → z = 3 |
Calculus Basics
| Formula name | Formula | Worked example |
|---|
| Power rule (derivative) | d/dx (xn) = n xn−1 | d/dx (x5) = 5x4 |
| Power rule (integral) | ∫ xn dx = xn+1 ÷ (n + 1) + C | ∫ x2 dx = x3 ÷ 3 + C |
| Product rule | (uv)′ = u′v + uv′ | (x sin x)′ = sin x + x cos x |
| Chain rule | d/dx f(g(x)) = f′(g(x)) × g′(x) | d/dx sin(2x) = 2 cos(2x) |
Finance & Interest
| Formula name | Formula | Worked example |
|---|
| Simple interest | I = P × r × t | P = 1000, r = 5%, t = 3 → I = 150 |
| Compound interest | A = P(1 + r ÷ n)nt | P = 1000, r = 5%, n = 1, t = 2 → A = 1102.50 |
| Daily interest | Id = P × (R ÷ 100) ÷ 365 | P = 10000, R = 12 → Id ≈ 3.29 |
| Loan EMI (monthly instalment) | EMI = P × r × (1 + r)n ÷ ((1 + r)n − 1) | P = 10000, r = 1%, n = 12 → EMI ≈ 888.49 |
| Return on investment (ROI) | ROI = ((G − C) ÷ C) × 100 | ((1500 − 1000) ÷ 1000) × 100 = 50% |
Physics & Everyday Science
| Formula name | Formula | Worked example |
|---|
| Speed | v = d ÷ t | 150 km ÷ 3 h = 50 km/h |
| Density | ρ = m ÷ V | 500 g ÷ 250 cm3 = 2 g/cm3 |
| Ohm's law | V = I × R | 2 A × 5 Ω = 10 V |
| Newton's second law | F = m × a | 10 kg × 3 m/s2 = 30 N |
| Kinetic energy | KE = ½mv2 | m = 2 kg, v = 3 m/s → KE = 9 J |
| Celsius to Fahrenheit | °F = °C × 9 ÷ 5 + 32 | 25°C → 77°F |
A–Z Index of Formulas
Jump to a letter, then pick a formula.
Formula FAQs
How do I use these formulas on a scientific calculator?
Type the expression exactly as written, using brackets around every numerator and denominator. For example, enter (45 ÷ 60) × 100 for a percentage, and use the π key for circle formulas.
Should I use DEG or RAD for trigonometry formulas?
Use DEG when the angle is in degrees and RAD when it is in radians. Mixing them up is the most common reason for a wrong sine, cosine or tangent answer.
How do I convert degrees to radians?
Multiply the angle in degrees by π ÷ 180. For example, 180° × π ÷ 180 = π radians, and 90° is π ÷ 2 ≈ 1.5708 radians.
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