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Math formulas

Arithmetic, algebra, geometry, trigonometry, statistics, probability and introductory calculus with examples.

Triangle

A = ½bh

b is the chosen base and h is the perpendicular distance to the opposite vertex. A sloping side is not usually the height.

Worked example

For b=10 cm and h=6 cm, A=30 cm². The result is half the corresponding rectangle’s area.

Circle

A = πr²; C = 2πr

r is the radius; diameter d=2r. Keep π until the final step to reduce rounding error.

Worked example

A circle with r=3 cm has A=9π≈28.27 cm² and C=6π≈18.85 cm.

Right triangle

a² + b² = c²

c is the hypotenuse opposite the right angle. This relationship applies only to right triangles.

Worked example

For perpendicular sides 3 and 4, c=√(9+16)=5. All lengths must share a unit.

Quadratic equation

x = (−b ± √(b² − 4ac)) / (2a)

For ax²+bx+c=0 with a≠0. The discriminant determines whether roots are distinct real, repeated real or complex.

Worked example

x²−5x+6=0 gives D=1 and x=(5±1)/2, so the roots are 3 and 2.

Arithmetic mean

Mean = Σx / n

Add all observations and divide by their count. This is an unweighted measure and can be influenced by extreme values.

Worked example

For 4, 7 and 10, mean=(4+7+10)/3=7. The mean need not occur in the original list.

Percentage change

Change = (new − original) / original × 100%

The original value must be nonzero. Positive change indicates an increase when the original is positive.

Worked example

A value rising from 80 to 100 increases by 25%. A return from 100 to 80 is a 20% decrease.

Simple probability

P(E) = favourable outcomes / total outcomes

Use this count-based formula when the outcomes are equally likely. Probabilities range from 0 to 1.

Worked example

A fair six-sided die has three even outcomes, so P(even)=3/6=1/2.

Exponents

aᵐaⁿ = aᵐ⁺ⁿ; (aᵐ)ⁿ = aᵐⁿ

These laws are straightforward for integer exponents. For negative powers, the base must be nonzero.

Worked example

2³×2⁴=2⁷=128. By contrast, 2³+2⁴=24, not 2⁷.

Algebraic identities

(a+b)² = a²+2ab+b²; (a−b)(a+b)=a²−b²

Expand brackets term by term. The middle term in a squared sum is essential.

Worked example

(10+3)²=100+60+9=169. These identities help with factorization and mental arithmetic.

Arithmetic progression

aₙ = a₁+(n−1)d; Sₙ = n[2a₁+(n−1)d]/2

a₁ is the first term, d the constant difference and n the positive integer number of terms.

Worked example

For 3,7,11,… the tenth term is 39 and the first ten terms sum to 210.

Geometric progression

aₙ = a₁qⁿ⁻¹; Sₙ = a₁(1−qⁿ)/(1−q)

q is a constant ratio. The sum formula assumes q≠1; when q=1, Sₙ=na₁.

Worked example

For 2,6,18,54 the ratio is 3 and the four-term sum is 80.

Trigonometry

sin θ = opposite/hypotenuse; cos θ = adjacent/hypotenuse; tan θ = opposite/adjacent

These side definitions apply to an acute angle in a right triangle. Check your calculator’s angle mode.

Worked example

For a 3–4–5 triangle with opposite side 3, sin θ=3/5 and cos θ=4/5.

Volume and surface area

Cuboid: V=lwh; surface area=2(lw+lh+wh)

Volume uses cubic units; surface area uses square units. A cylinder has V=πr²h.

Worked example

A 2 cm × 3 cm × 4 cm cuboid has volume 24 cm³ and surface area 52 cm².

Coordinate geometry

Distance = √[(x₂−x₁)²+(y₂−y₁)²]

This is the straight-line distance in a Cartesian plane. Midpoint coordinates are the averages of corresponding coordinates.

Worked example

Between (1,2) and (4,6), distance is 5 and midpoint is (2.5,4).

Logarithms

logₐ(xy)=logₐx+logₐy

For real logarithms require a>0, a≠1 and positive arguments. A logarithm gives the exponent needed to obtain a number.

Worked example

log₁₀1000=3 because 10³=1000. log(x+y) does not equal log x + log y.

Number systems and divisibility

Natural numbers are counting numbers; whole numbers also include zero. Integers include negative whole numbers. Rational numbers can be written p/q with integers p and q, q ≠ 0. Irrational numbers cannot, for example √2 and π. Real numbers combine rational and irrational numbers. Prime numbers have exactly two positive divisors; 1 is neither prime nor composite.

Divisor	Test	Example
2	Last digit is even	348
3	Digit sum divisible by 3	348: sum 15
4	Last two digits divisible by 4	1316
5	Last digit 0 or 5	735
6	Divisible by 2 and 3	348
8	Last three digits divisible by 8	5120
9	Digit sum divisible by 9	729: sum 18
11	Alternating digit sums differ by a multiple of 11	121: (1+1)−2=0

For positive integers a and b: HCF(a,b) × LCM(a,b) = a × b. Prime factorisation uses the smallest exponent in HCF and largest in LCM. For 12=2²×3 and 18=2×3², HCF=6 and LCM=36.

Fractions, decimals and order of operations

a/b + c/d = (ad+bc)/bd; (a/b) ÷ (c/d) = ad/bc

Denominators must be non-zero, and c must also be non-zero when dividing. Evaluate brackets, powers, multiplication and division left to right, then addition and subtraction left to right. A terminating or repeating decimal is rational.

Worked example

3/4 + 2/3 = (9+8)/12 = 17/12. Also 0.125 = 125/1000 = 1/8. In 18 ÷ 3 × 2, work left to right to obtain 12.

Common mistake: Adding denominators when adding fractions, or assuming multiplication always precedes division.

Ratios, proportion and unitary method

a:b = a/b; a/b = c/d ⇒ ad = bc

Compare quantities in the same units. In direct proportion y=kx; in inverse proportion xy=k. The constant k is fixed only under the stated conditions. Divide a total in ratio a:b using a/(a+b) and b/(a+b).

Worked example

Split ₹840 in ratio 3:4: each part is ₹120, giving ₹360 and ₹480. Six identical workers taking 10 days implies 60 worker-days; 15 workers need 4 days at unchanged productivity.

Common mistake: Applying inverse proportion when the variables actually increase together.

Profit, loss, discount and interest

Profit% = 100(SP−CP)/CP; discount% = 100(MP−SP)/MP

CP is cost price, SP selling price and MP marked price. Simple interest I=Prt/100 with annual rate r% and time t in years. Compound amount A=P(1+r/(100n))^(nt), with n compounding periods per year and a fixed nominal annual rate.

Worked example

An item costing ₹800 and sold for ₹920 earns ₹120, or 15% profit. ₹1000 compounded annually at 10% becomes ₹1210 after two years; interest is ₹210.

Common mistake: Using selling price as the base for profit percentage; confusing profit margin with markup.

Speed, distance, time and work

distance = speed × time; combined work rate = 1/a + 1/b

Use average speed = total distance / total elapsed time. For opposite directions, relative speeds add; for the same direction, take their difference. If A completes one job in a days, the constant daily rate is 1/a.

Worked example

A and B take 12 and 18 days independently: rate=5/36 jobs/day, so time=7.2 days. A 60 km trip at 30 km/h and return at 60 km/h averages 120/3=40 km/h.

Common mistake: Taking the arithmetic mean of speeds for equal-distance journeys.

Plane shapes and angle facts

Shape or fact	Formula	Meaning
Rectangle	A=lw; P=2(l+w)	l length, w width
Square	A=a²; P=4a; diagonal=a√2	a side
Parallelogram	A=bh	h perpendicular height
Trapezium	A=½(a+b)h	a,b parallel sides
Rhombus	A=½d₁d₂	diagonals d₁,d₂
Triangle	A=√[s(s−a)(s−b)(s−c)]	s=(a+b+c)/2
Regular n-gon	Interior-angle sum=(n−2)180°	Each angle=(n−2)180°/n
Sector	A=(θ/360°)πr²	θ in degrees
Arc	length=(θ/360°)2πr	θ in degrees

Angles on a straight line sum to 180°; around a point they sum to 360°. Triangle interior angles sum to 180°. The angle subtended by a diameter at the circumference is 90°. For a trapezium with parallel sides 8 and 12 cm and height 5 cm, area=50 cm².

Trigonometric values and identities

Angle	sin θ	cos θ	tan θ
0°	0	1	0
30°	1/2	√3/2	1/√3
45°	1/√2	1/√2	1
60°	√3/2	1/2	√3
90°	1	0	Undefined

sin²θ+cos²θ=1; tanθ=sinθ/cosθ when defined. 1+tan²θ=sec²θ; 1+cot²θ=cosec²θ. sin(A±B)=sinA cosB ± cosA sinB; cos(A±B)=cosA cosB ∓ sinA sinB. sin2A=2sinA cosA and cos2A=cos²A−sin²A. A full turn is 2π radians; radians=degrees×π/180. For opposite side 3 and hypotenuse 5 in a right triangle, sinθ=3/5 and adjacent side=4.

Permutations, combinations and binomial expansion

nPr = n!/(n−r)!; nCr = n!/[r!(n−r)!]

Use permutations when order matters and combinations when it does not, for r selections from n distinct items without replacement. Define 0!=1. For nonnegative integer n, (a+b)^n = Σ from r=0 to n of nCr a^(n−r)b^r.

Worked example

Choose 3 people from 5: 5C3=10. Award three distinct medals among 5 people: 5P3=60. (x+1)³=x³+3x²+3x+1.

Common mistake: Using permutations for an unordered committee. Repeated objects and replacement need different rules.

Conditional probability and independence

P(A∪B)=P(A)+P(B)−P(A∩B); P(A|B)=P(A∩B)/P(B)

Conditional probability requires P(B)>0. Independence means P(A∩B)=P(A)P(B). Mutually exclusive events cannot occur together. The complement has probability 1−P(A).

Worked example

On a fair die A=even, B=greater than 4. P(A∪B)=3/6+2/6−1/6=4/6. Given B, only 5 and 6 remain, so P(A|B)=1/2.

Common mistake: Treating mutually exclusive events with positive probabilities as independent.

Median, mode, variance and weighted mean

Weighted mean = Σ(wx)/Σw; population variance = Σ(x−μ)²/N

Sort data for the median; for even N average the middle two values. Mode is the most frequent value and need not be unique. Standard deviation is the square root of variance. Sample variance commonly divides by n−1 to estimate population variance.

Worked example

For 2,4,4,6: mean=4, median=4, mode=4. Population variance=8/4=2. Weights 2 and 3 on marks 70 and 80 give 76.

Common mistake: Confusing sample and population variance, or forgetting weights.

Sets, functions and domain

n(A∪B)=n(A)+n(B)−n(A∩B)

A set contains distinct elements. A function assigns exactly one output to each input in its domain. Exclude zero denominators, negative arguments under real even roots and non-positive logarithm arguments.

Worked example

For f(x)=√(x−2)/(x−3), the real domain is x≥2 excluding 3. If 18 students study Hindi, 15 English and 8 both, 25 study at least one.

Common mistake: Cancelling an expression without retaining its domain restrictions.

Limits and derivatives

d(xⁿ)/dx=nx^(n−1); (uv)′=u′v+uv′; (u/v)′=(u′v−uv′)/v²

A derivative is an instantaneous rate of change. Chain rule: d[f(g(x))]/dx=f′(g(x))g′(x). Derivatives of sin x, cos x, eˣ and ln x are cos x, −sin x, eˣ and 1/x respectively, on their domains. Trigonometric derivative formulas assume radians.

Worked example

For y=(3x+1)², dy/dx=6(3x+1). The standard limit sin x/x→1 as x→0 also assumes radians.

Common mistake: Forgetting the inner derivative.

Integration and definite area

∫xⁿ dx=x^(n+1)/(n+1)+C, n≠−1; ∫1/x dx=ln|x|+C

An indefinite integral gives a family of antiderivatives. For continuous f, ∫ₐᵇf(x)dx=F(b)−F(a) gives signed accumulation. Geometric area can require splitting at zeros and taking absolute contributions.

Worked example

∫₀²3x² dx=[x³]₀²=8; ∫cos x dx=sin x+C. A function below the x-axis contributes negatively to a definite integral.

Common mistake: Omitting C for indefinite integrals or assuming signed area must be positive.

Matrices and determinants

For A=[[a,b],[c,d]], det A=ad−bc

Matrix addition needs equal dimensions. An m×n matrix multiplies an n×p matrix to give m×p using row-by-column products. If det A≠0, A⁻¹=(1/det A)[[d,−b],[−c,a]]. Matrix multiplication generally does not commute.

Worked example

A=[[2,1],[3,2]] has determinant 1 and inverse [[2,−1],[−3,2]]. Their product is the identity.

Common mistake: Multiplying corresponding entries instead of rows by columns.

Vectors and complex numbers

a·b=|a||b|cosθ; z=a+bi; i²=−1

For Cartesian vectors a·b=aₓbₓ+aᵧbᵧ+a_zb_z. Orthogonal nonzero vectors have dot product zero. A complex number has modulus √(a²+b²) and conjugate a−bi. Use the conjugate to divide complex numbers.

Worked example

(1,2)·(3,−1)=1. (2+3i)(2−3i)=13 and |2+3i|=√13.

Common mistake: Treating i² as +1 or confusing dot and cross products.

Practice with answers

Find the LCM of 18 and 24.

18=2×3² and 24=2³×3, giving LCM=72.

A price rises 20% then falls 20%. Is it unchanged?

100×1.2×0.8=96; net decrease 4%.

Solve x²−5x+6=0.

(x−2)(x−3)=0, so x=2 or 3.

Area of a 90° sector of radius 4 cm?

(90/360)π×16=4π cm².

Differentiate x²sin x.

2x sin x+x²cos x by the product rule.

How many unordered pairs from 6 students?

6C2=15.

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