Maths made clear, practical & easy.

Subtraction

\(a-b\)

Find the difference between two quantities.

Multiplication

a\times b

Repeated addition of equal groups.

Division

a\div b

Share a quantity into equal groups; divisor must not be zero.

Place Value

d\times 10^n

Value of a digit depends on its position.

Money Conversion

\text{paise}=100\times\text{rupees}

One rupee equals one hundred paise.

Total Cost

\text{price}\times\text{quantity}

Multiply price per item by number of items.

Learn formula →Arithmetic

Remainder Relation

\(a=bq+r\)

Dividend equals divisor times quotient plus remainder; 0 ≤ r < |b|.

Even Number

\(n=2k\)

An integer is even when divisible by 2.

Learn formula →Arithmetic

Odd Number

\(n=2k+1\)

An integer is odd when it is one more than an even integer.

Learn formula →Measurement

Centimetres to Metres

\(m=cm/100\)

Convert centimetres to metres.

Open calculator →Measurement

Metres to Kilometres

\(km=m/1000\)

Convert metres to kilometres.

Open calculator →Measurement

Grams to Kilograms

\(kg=g/1000\)

Convert grams to kilograms.

Open calculator →Measurement

Minutes to Hours

\(h=m/60\)

Convert minutes to hours.

Rectangle Perimeter

\(P=2(l+b)\)

Distance around a rectangle.

Square Perimeter

\(P=4a\)

Four equal sides of a square.

Rectangle Area

A=l\times b

Area enclosed by a rectangle.

Square Area

\(A=a^2\)

Area enclosed by a square.

Fraction of a Quantity

\frac{a}{b}\times N

Find a fractional part of a whole number.

Fraction Addition

\frac ab+\frac cd=\frac{ad+bc}{bd}

Add two fractions using a common denominator.

Fraction Multiplication

\frac ab\times\frac cd=\frac{ac}{bd}

Multiply the numerators and denominators.

Average (Mean)

\bar x=\frac{\sum x_i}{n}

Add values and divide by number of values.

Percentage

p=\frac{\text{part}}{\text{whole}}\times100

Express part of a whole out of one hundred.

Percent of a Number

x\times\frac{p}{100}

Calculate a given percentage of a number.

Ratio Simplification

a:b=\frac{a}{g}:\frac{b}{g}

Divide both terms by their greatest common divisor.

HCF of Two Integers

\operatorname{HCF}(a,b)

Greatest positive integer dividing both integers.

LCM of Two Integers

\operatorname{LCM}(a,b)=\frac{|ab|}{\operatorname{HCF}(a,b)}

Smallest nonnegative common multiple; both nonzero for positive LCM.

Triangle Area

A=\frac12 bh

Half the product of base and perpendicular height.

Parallelogram Area

\(A=bh\)

Base times perpendicular height.

Circle Circumference

C=2\pi r

Distance around a circle of radius r.

Circle Area

A=\pi r^2

Area inside a circle of radius r.

Simple Interest

SI=\frac{PRT}{100}

Interest on principal P at annual rate R% for T years.

Profit

\text{Profit}=SP-CP

Selling price minus cost price.

Profit Percentage

\frac{SP-CP}{CP}\times100

Profit relative to cost price; CP must be positive.

Discount

D=\frac{MP\times d}{100}

Discount on marked price at rate d%.

Speed

v=\frac{d}{t}

Speed equals distance divided by elapsed time.

Distance

\(d=vt\)

Distance at constant speed.

Time

t=\frac{d}{v}

Travel time at constant nonzero speed.

Cube Volume

\(V=a^3\)

Volume of a cube with side a.

Cuboid Volume

\(V=lbh\)

Volume of a rectangular box.

Cube Total Surface Area

\(S=6a^2\)

Six equal square faces.

Cuboid Total Surface Area

\(S=2(lb+bh+hl)\)

Sum of areas of six rectangular faces.

Compound Amount

A=P(1+\frac{r}{100})^t

Amount with annual compounding.

Square of a Sum

\((a+b)^2=a^2+2ab+b^2\)

Expand the square of a binomial.

Square of a Difference

\((a-b)^2=a^2-2ab+b^2\)

Expand the square of a difference.

Difference of Squares

\(a^2-b^2=(a-b)(a+b)\)

Factor a difference of two squares.

Product Identity

\((x+a)(x+b)=x^2+(a+b)x+ab\)

Expand two linear factors.

Linear Equation

ax+b=c\Rightarrow x=\frac{c-b}{a}

Solve an equation in one variable when a ≠ 0.

Exponent Product

a^m a^n=a^{m+n}

Same base: add exponents.

Exponent Quotient

\frac{a^m}{a^n}=a^{m-n}

Same nonzero base: subtract exponents.

Power of a Power

(a^m)^n=a^{mn}

Multiply exponents.

Learn formula →Geometry

Trapezium Area

A=\frac12(a+b)h

Average of parallel sides times perpendicular height.

Rhombus Area

A=\frac12d_1d_2

Half product of diagonals.

Cylinder Volume

V=\pi r^2h

Circular base area times height.

Cylinder Curved Surface Area

CSA=2\pi rh

Area of curved part of a right circular cylinder.

Cone Volume

V=\frac13\pi r^2h

One third of matching cylinder volume.

Sphere Volume

V=\frac43\pi r^3

Volume of a sphere.

Sphere Surface Area

S=4\pi r^2

Surface area of a sphere.

Pythagoras Theorem

\(a^2+b^2=c^2\)

Right triangle legs a,b and hypotenuse c.

Heron Formula

A=\sqrt{s(s-a)(s-b)(s-c)}

Triangle area from sides; s=(a+b+c)/2.

Quadratic Discriminant

\(D=b^2-4ac\)

Determines nature of quadratic roots.

Quadratic Roots

x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}

Find roots of ax²+bx+c=0, a ≠ 0.

Sum of Quadratic Roots

\alpha+\beta=-\frac ba

Sum of roots of ax²+bx+c=0.

Product of Quadratic Roots

\alpha\beta=\frac ca

Product of roots of ax²+bx+c=0.

Arithmetic Progression nth Term

\(a_n=a+(n-1)d\)

Find the nth term of an arithmetic progression.

Arithmetic Progression Sum

S_n=\frac n2[2a+(n-1)d]

Sum of first n terms of an arithmetic progression.

Distance Between Points

d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}

Distance between two points in a plane.

Midpoint

M=(\frac{x_1+x_2}2,\frac{y_1+y_2}2)

Midpoint of a line segment.

Slope

m=\frac{y_2-y_1}{x_2-x_1}

Rate of change of y with x for nonvertical line.

Section Formula

P=(\frac{mx_2+nx_1}{m+n},\frac{my_2+ny_1}{m+n})

Point dividing segment internally in ratio m:n.

Sine Ratio

\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}

Right triangle definition for acute angle θ.

Cosine Ratio

\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}

Right triangle definition for acute angle θ.

Tangent Ratio

\tan\theta=\frac{\text{opposite}}{\text{adjacent}}

Right triangle definition when adjacent is nonzero.

Pythagorean Identity

\sin^2\theta+\cos^2\theta=1

Fundamental identity for all real angles.

Tangent Identity

1+\tan^2\theta=\sec^2\theta

Where tangent and secant are defined.

Learn formula →Geometry

Sector Area

A=\frac{\theta}{360}\pi r^2

Area of circular sector for angle θ in degrees.

Arc Length

s=\frac{\theta}{360}2\pi r

Arc length for central angle θ in degrees.

Median (Odd n)

x_{(n+1)/2}

Central ordered observation for odd sample size.

Mode

\text{mode}=\text{most frequent value}

Most frequent value(s) in data set.

Classical Probability

P(E)=\frac{\text{favourable outcomes}}{\text{equally likely outcomes}}

Valid for equally likely outcomes.

Population Variance

\sigma^2=\frac{\sum(x_i-\mu)^2}{N}

Population mean squared deviation.

Population Standard Deviation

\sigma=\sqrt{\frac{\sum(x_i-\mu)^2}{N}}

Square root of population variance.

Geometric Progression nth Term

a_n=ar^{n-1}

Find nth term of geometric sequence.

Geometric Progression Sum

S_n=a\frac{r^n-1}{r-1}

Sum of n geometric terms when r ≠ 1; for r=1, S=na.

Permutations

{}^nP_r=\frac{n!}{(n-r)!}

Ordered selections of r from n distinct objects.

Combinations

{}^nC_r=\frac{n!}{r!(n-r)!}

Unordered selections of r from n distinct objects.

Binomial General Term

T_{r+1}={}^{n}C_r a^{n-r}b^r

General term in expansion of (a+b)^n.

Sine Rule

\frac a{\sin A}=\frac b{\sin B}=\frac c{\sin C}

Relates triangle sides to their opposite angles.

Cosine Rule

c^2=a^2+b^2-2ab\cos C

Find third side using included angle.

Learn formula →Coordinate Geometry

Circle Equation

\((x-h)^2+(y-k)^2=r^2\)

Circle with centre (h,k) and radius r.

Straight Line Slope Intercept

\(y=mx+c\)

Line with slope m and y-intercept c.

Learn formula →Coordinate Geometry

Parabola Standard Equation

\(y^2=4ax\)

Right/left opening parabola based on sign of a.

Learn formula →Coordinate Geometry

Ellipse Standard Equation

\frac{x^2}{a^2}+\frac{y^2}{b^2}=1

Ellipse centered at origin for a,b>0.

Power Rule for Derivatives

\frac{d}{dx}x^n=nx^{n-1}

Differentiate a power of x.

Derivative of Sine

\frac{d}{dx}\sin x=\cos x

Angles in radians.

Derivative of Cosine

\frac{d}{dx}\cos x=-\sin x

Angles in radians.

Product Rule

(fg)\prime=f\prime g+fg\prime

Differentiate product of two differentiable functions.

Quotient Rule

(f/g)\prime=\frac{gf\prime-fg\prime}{g^2}

Differentiate quotient when g ≠ 0.

Chain Rule

\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}

Differentiate a composition of functions.

Integral Power Rule

\int x^n dx=\frac{x^{n+1}}{n+1}+C

For n ≠ −1; indefinite integral.

Logarithmic Integral

\int\frac1x dx=\ln|x|+C

Antiderivative of 1/x for x ≠ 0.

Open calculator →Matrices

Fundamental Theorem of Calculus

\int_a^b f(x)dx=F(b)-F(a)

F is an antiderivative of continuous f.

Learn formula →Vectors

Dot Product

\vec a\cdot\vec b=a_1b_1+a_2b_2+a_3b_3

Scalar product of vectors in 3D.

Open calculator →Vectors

Cross Product Magnitude

|\vec a\times\vec b|=|\vec a||\vec b|\sin\theta

Magnitude of vector product.

Learn formula →Vectors

Vector Magnitude

|\vec a|=\sqrt{a_1^2+a_2^2+a_3^2}

Length of 3D vector.

2 by 2 Determinant

\det(A)=ad-bc

Determinant of matrix [[a,b],[c,d]].

Open calculator →Matrices

2 by 2 Matrix Inverse

A^{-1}=\frac1{ad-bc}\begin{bmatrix}d&-b\\-c&a\end{bmatrix}

Inverse exists when determinant is not zero.

Learn formula →Probability

Complement Rule

\(P(A^c)=1-P(A)\)

Probability that A does not occur.

Addition Rule

P(A\cup B)=P(A)+P(B)-P(A\cap B)

Probability of A or B.

Conditional Probability

P(A|B)=\frac{P(A\cap B)}{P(B)}

When P(B)>0.

Bayes Theorem

P(A|B)=\frac{P(B|A)P(A)}{P(B)}

Bayes theorem requires P(B)>0.

Degree to Radian

rad=deg\times\frac\pi{180}

Convert angle degrees to radians.