O Level / IGCSE Mathematics formulas

Every chapter of O Level / IGCSE Mathematics on one page: the 446 formulas, definitions and facts to remember, in syllabus order. Use it for a last look before a test, then check yourself.

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Number

Types of number

Natural numbers are the counting numbers 1, 2, 3, …; integers are …, −2, −1, 0, 1, 2, …
A prime has exactly two factors. 1 is not prime and 2 is the only even prime.
To test whether a number is prime, try dividing by the primes up to its square root.
HCF = product of the common prime factors, each with its lowest power.
LCM = product of every prime factor that appears, each with its highest power.
In a square number every prime factor has an even power.
reciprocal of n = 1 ÷ n, so the reciprocal of a/b is b/a

Sets

A ∩ B: in both A and B. A ∪ B: in A or B or both.
A′: in ξ but not in A, so n(A′) = n(ξ) − n(A).
n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
number in neither set = n(ξ) − n(A ∪ B)
number in A only = n(A) − n(A ∩ B); this region is the set A ∩ B′
A ⊆ B means every element of A is also in B; A ⊂ B (proper subset) means this and A is not equal to B.
∅ is the empty set; x ∈ A means x is an element of A.

Powers and roots

a² = a × a and a³ = a × a × a
Squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225
Cubes: 1, 8, 27, 64, 125 and 10³ = 1000
(−a)² is positive; (−a)³ is negative, so ∛(−8) = −2
√(a/b) = √a ÷ √b and ∛(a/b) = ∛a ÷ ∛b
For decimals use fractions: √0.25 = √(25/100) = 5/10 = 0.5
To place a root between whole numbers, find the squares on each side: 36 < 40 < 49, so √40 is between 6 and 7.

Fractions, decimals and percentages

Fraction to decimal: numerator ÷ denominator. Decimal to percentage: × 100.
Add or subtract: change to a common denominator first, then add or subtract the numerators.
Multiply: (a/b) × (c/d) = (a × c)/(b × d)
Divide: (a/b) ÷ (c/d) = (a/b) × (d/c)
Mixed number to improper fraction: 2 1/3 = (2 × 3 + 1)/3 = 7/3
A fraction in lowest terms terminates only if its denominator has no prime factors other than 2 and 5.
Recurring decimal: let x be the decimal, multiply by 10, 100, … to line up the repeating part, then subtract.

Ordering

a > b means a is greater than b; a < b means a is less than b; ≥ and ≤ include “equal to”.
Change fractions and percentages to decimals before comparing.
Write the decimals to the same number of places: 0.7 = 0.700, so 0.7 > 0.69.
For negative numbers the order reverses: 5 > 3 but −5 < −3.
For a number between 0 and 1, squaring makes it smaller and the square root makes it larger.
Dividing by a number between 0 and 1 gives a larger answer.

The four operations

Order: Brackets, Indices (powers), Division and Multiplication, Addition and Subtraction.
(+) × (+) = (+), (−) × (−) = (+), (+) × (−) = (−); the same pattern holds for division.
Subtracting a negative is the same as adding: a − (−b) = a + b
(−a)² is positive and (−a)³ is negative.
Multiplying decimals: multiply as whole numbers, then count the decimal places: 0.3 × 0.02 = 0.006.
Dividing by a decimal: multiply both numbers by 10 or 100 first: 4.8 ÷ 0.06 = 480 ÷ 6 = 80.

Indices I

a^m × aⁿ = a^{m + n}
a^m ÷ aⁿ = a^{m − n}
(a^m)ⁿ = a^{m × n}
a⁰ = 1 (for a not equal to 0)
a⁻ⁿ = 1 ÷ aⁿ, and (a/b)⁻ⁿ = (b/a)ⁿ
a^1/n = the nth root of a, so 9^1/2 = 3 and 8^1/3 = 2
a^m/n = (nth root of a)^m

Standard form

Standard form is A × 10ⁿ with 1 ≤ A < 10 and n an integer.
Large number: n is positive. Number less than 1: n is negative.
Multiply: multiply the A values and add the indices.
Divide: divide the A values and subtract the indices.
Add or subtract: make the powers of ten the same first, or change to ordinary numbers.
Adjusting: 24 × 10³ = 2.4 × 10⁴ and 0.5 × 10⁸ = 5 × 10⁷.
(A × 10ⁿ)² = A² × 10²ⁿ

Estimation

Significant figures are counted from the first non-zero digit: 0.004 07 has 3 significant figures.
Next digit 5 or more: round up. Next digit 4 or less: leave the digit as it is.
Keep the size of the number: 48 651 to 2 significant figures is 49 000, not 49.
Keep a final zero if it is significant: 2.596 to 3 significant figures is 2.60.
Estimating: round each number to 1 significant figure, then calculate.
Dividing by a number less than 1 makes the answer bigger: 240 ÷ 0.4 = 600.
Round up when a whole number of containers or vehicles must hold everything.

Limits of accuracy

lower bound = value − half the rounding unit; upper bound = value + half the rounding unit
Nearest 10: ± 5. Nearest whole number: ± 0.5. 1 decimal place: ± 0.05. Nearest 5: ± 2.5.
Error interval: lower bound ≤ x < upper bound
Upper bound of a + b or a × b: use both upper bounds.
Upper bound of a − b: upper a − lower b. Lower bound of a − b: lower a − upper b.
Upper bound of a ÷ b: upper a ÷ lower b. Lower bound of a ÷ b: lower a ÷ upper b.

Ratio and proportion

one part = given amount ÷ number of parts it stands for
Sharing in the ratio a : b: the shares are a/(a + b) and b/(a + b) of the total.
Simplify by dividing every part by the same number, with all quantities in the same units.
Direct proportion: find the value for one, then multiply (the unitary method).
Inverse proportion: the product stays the same, so workers × days is constant.
Scale 1 : n: actual distance = map distance × n. 1 km = 100 000 cm.
To join a : b and b : c, make the b values equal first.

Rates

speed = distance ÷ time; distance = speed × time; time = distance ÷ speed
average speed = total distance ÷ total time
density = mass ÷ volume
population density = number of people ÷ area
km/h to m/s: divide by 3.6. m/s to km/h: multiply by 3.6.
minutes to hours: divide by 60, so 45 minutes = 0.75 hour and 2 h 20 min = 2 1/3 h
A rate per second from a rate per minute: divide by 60.

Percentages

percentage of a quantity = (percentage ÷ 100) × quantity
A as a percentage of B = (A ÷ B) × 100, with A and B in the same units
percentage change = (change ÷ original value) × 100
simple interest = principal × rate × time ÷ 100 (I = PRT/100)
compound interest: total amount = P × (1 + r/100)ⁿ after n years
interest earned = total amount − amount invested
reverse percentage: original amount = new amount ÷ multiplier

Using a calculator

Order: brackets, then powers and roots, then × and ÷, then + and −
For a fraction, put brackets round the whole top and round the whole bottom: (a + b) ÷ (c × d)
A display of 3.2E5 means 3.2 × 10⁵ = 320 000; 3.2E−5 means 3.2 × 10⁻⁵ = 0.000032
Money: a display of 7.4 means $7.40
Time: 2.3 hours = 2 hours and 0.3 × 60 minutes = 2 hours 18 minutes
For sin, cos and tan of angles in degrees, the calculator must be in degree mode
To square a negative number, use brackets: (−3)² = 9

Time

1 minute = 60 seconds; 1 hour = 60 minutes = 3600 seconds
1 day = 24 hours; 1 week = 7 days; 1 year = 12 months = 365 days (366 in a leap year)
12-hour to 24-hour: add 12 to pm times from 1:00 pm to 11:59 pm, so 7:05 pm = 19:05
12:00 midnight = 00:00 and 12:00 noon = 12:00
minutes to hours: divide by 60, so 45 minutes = 0.75 hours
decimal hours to minutes: multiply the decimal part by 60, so 0.4 hours = 24 minutes
Time zones: find the arrival time in the starting place's time first, then add or subtract the time difference

Money

total cost = price of one item × number of items
change = amount paid − total cost
profit = selling price − cost price (a negative answer is a loss)
If $1 = k units of another currency: dollars × k = other currency, and other currency ÷ k = dollars
best value = the lowest price for the same quantity (price ÷ quantity)
time-and-a-half = 1.5 × the normal hourly rate; double time = 2 × the normal rate
instalment price = deposit + (number of payments × each payment)

Exponential growth and decay

growth multiplier = 1 + r/100, so 5% growth gives 1.05
decay multiplier = 1 − r/100, so a 12% decrease gives 0.88
value after n periods = starting value × multiplierⁿ
number of periods n = total time ÷ length of one period
In y = A × bⁿ, A is the starting value; b > 1 means growth and 0 < b < 1 means decay
To go back n periods in time, divide by multiplierⁿ

Surds

√(ab) = √a × √b and √(a/b) = √a ÷ √b
√a × √a = a
Like surds: p√a + q√a = (p + q)√a, for example 3√2 + 5√2 = 8√2
√a + √b is not equal to √(a + b)
To rationalise k/√a, multiply top and bottom by √a: k/√a = k√a/a
(a + √b)(a − √b) = a² − b
To rationalise k/(a + √b), multiply top and bottom by (a − √b); for k/(a − √b), use (a + √b)

Algebra and graphs

Introduction to algebra

3x means 3 × x, xy means x × y, and x/y means x ÷ y
x² means x × x; 3x² means 3 × (x²), so square first, then multiply by 3
(−a)² = a², for example (−4)² = 16
Subtracting a negative: p − (−q) = p + q
negative × negative = positive; negative × positive = negative
"4 less than n" is n − 4; "3 more than n" is n + 3; "twice n" is 2n

Algebraic manipulation

Only like terms can be added or subtracted: 5a + 2a = 7a, but 5a + 2b stays as it is
a(b + c) = ab + ac; a minus sign in front of a bracket changes every sign inside: −(2a − 7b) = −2a + 7b
(a + b)(c + d) = ac + ad + bc + bd
(a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b²
Difference of two squares: a² − b² = (a + b)(a − b)
x² + bx + c = (x + p)(x + q), where p + q = b and p × q = c
Completing the square: x² + bx + c = (x + b/2)² − (b/2)² + c

Algebraic fractions

Simplify: factorise top and bottom, then cancel common factors, for example (x² − 4)/(x² + 2x) = (x − 2)(x + 2)/(x(x + 2)) = (x − 2)/x
a/b + c/d = (ad + bc)/(bd)
a/b − c/d = (ad − bc)/(bd)
(a/b) × (c/d) = (ac)/(bd)
(a/b) ÷ (c/d) = (a/b) × (d/c) = (ad)/(bc)
When you subtract, keep the second top in brackets so that every sign changes

Indices II

a^m × aⁿ = a^{m + n}
a^m ÷ aⁿ = a^{m − n}
(a^m)ⁿ = a^mn and (ab)ⁿ = aⁿbⁿ
a⁰ = 1 (a not zero)
a⁻ⁿ = 1/aⁿ
a^1/n = the nth root of a, so a^1/2 = √a
a^m/n = (nth root of a)^m, for example 8^2/3 = 2² = 4

Equations

Linear equation: expand brackets, collect the letter on one side and the numbers on the other, then divide
If a/b = c/d then ad = bc (cross-multiply)
Simultaneous equations: same signs subtract, different signs add, to eliminate one letter
If (x − p)(x − q) = 0 then x = p or x = q
Quadratic formula for ax² + bx + c = 0: x = (−b ± √(b² − 4ac))/(2a)
Changing the subject when the letter appears twice: collect its terms on one side, factorise, then divide
Always check by putting your answer back into the original equation

Inequalities

< less than; > greater than; ≤ less than or equal to; ≥ greater than or equal to
Multiplying or dividing both sides by a negative number reverses the sign: −2x > 6 gives x < −3
Number line: open circle for < and >, filled circle for ≤ and ≥
Graphs: broken (dashed) line for < and >, solid line for ≤ and ≥
y > mx + c is the region above the line y = mx + c; y < mx + c is the region below it
x > k is the region to the right of the vertical line x = k; x < k is to the left
To check a region, put the coordinates of a test point such as (0, 0) into the inequality

Sequences

Linear sequence with common difference d: nth term = dn + (first term − d)
Quadratic sequence: the second difference is constant, and the coefficient of n² = second difference ÷ 2
Geometric sequence with first term a and multiplier r: nth term = a × r^{n − 1}
Square numbers 1, 4, 9, 16, 25 have nth term n²; cube numbers 1, 8, 27, 64 have nth term n³
Triangular numbers 1, 3, 6, 10, 15 have nth term n(n + 1)/2
To find which term has a given value, put the nth term equal to that value and solve for n

Proportion

y directly proportional to x: y = kx
y directly proportional to x²: y = kx² (and y = kx³, y = k√x in the same way)
y inversely proportional to x: y = k/x
y inversely proportional to x²: y = k/x² (and y = k/√x in the same way)
The symbol ∝ means "is proportional to"
If y = kx², doubling x makes y 4 times as large; if y = k/x², doubling x makes y one quarter as large

Graphs in practical situations

speed = distance ÷ time = gradient of a distance–time graph
average speed = total distance ÷ total time
acceleration = change in speed ÷ time taken = gradient of a speed–time graph (unit m/s²)
distance travelled = area under a speed–time graph
area of a triangle = ½ × base × height; area of a trapezium = ½ × (sum of the parallel sides) × distance between them
A conversion graph is a straight line: read across from one axis and down to the other
A fixed charge plus a rate per hour gives a straight line that starts at the fixed charge on the vertical axis

Graphs of functions

Roots of f(x) = 0 are the x-coordinates where y = f(x) crosses the x-axis
To solve f(x) = k, draw the horizontal line y = k and read the x-coordinates of the intersections
To solve another equation, rearrange it so the drawn function is on one side; the other side is the line to draw
y = a/x has two branches and never touches the axes: there is no value at x = 0
y = b^x passes through (0, 1); it shows growth when b > 1 and decay when 0 < b < 1
For y = a × b^x the starting value (x = 0) is a, and y is multiplied by b each time x goes up by 1
gradient of a tangent = change in y ÷ change in x, using two points far apart on the tangent

Sketching curves

y-intercept: put x = 0 (for y = ax² + bx + c it is c)
x-intercepts (roots): put y = 0 and solve, usually by factorising
a > 0 gives a minimum point (U shape); a < 0 gives a maximum point
Line of symmetry of y = ax² + bx + c: x = −b/(2a), which is also halfway between the roots
y = (x − p)² + q has its turning point at (p, q)
y = (x − p)(x − q) crosses the x-axis at x = p and x = q
A quadratic with roots p and q is y = k(x − p)(x − q); find k from another point on the curve

Functions

f(a) means replace x with a in the formula
To solve f(x) = k, put the formula equal to k and solve for x
To find f⁻¹(x): write y = f(x), rearrange to make x the subject, then replace y with x
fg(x) = f(g(x)): apply g first, then f
f⁻¹(f(x)) = x, which is a quick check of your inverse
For the range, check the end values of the domain and any turning point between them

Coordinate geometry

Coordinates

(x, y): x is across, y is up or down; x always comes first
Points on the x-axis have the form (a, 0); points on the y-axis have the form (0, b)
The line x = a is vertical; the line y = b is horizontal; they meet at (a, b)
Length of a horizontal line = difference of the x-coordinates; length of a vertical line = difference of the y-coordinates
In a parallelogram ABCD, the movement from A to B is the same as the movement from D to C
A point the same distance from A and B lies on the line through the midpoint of AB at right angles to AB

Drawing linear graphs

A point lies on a line if its coordinates make the equation true
y-intercept: put x = 0; x-intercept: put y = 0
y = mx + c crosses the y-axis at (0, c)
y = mx (no constant term) passes through the origin
x = a is a vertical line through (a, 0); y = b is a horizontal line through (0, b)
Two lines cross at the point whose coordinates fit both equations (the solution of the simultaneous equations)

Gradient of linear graphs

gradient = change in y ÷ change in x (rise ÷ run)
For points (x₁, y₁) and (x₂, y₂): m = (y₂ − y₁)/(x₂ − x₁)
In y = mx + c the gradient is m; rearrange to this form first if needed
ax + by = c has gradient −a/b
Horizontal line (y = b): gradient 0; vertical line (x = a): gradient undefined
Three points lie on one straight line if the gradient between each pair is the same

Length and midpoint

Length of the segment from (x₁, y₁) to (x₂, y₂) = √[(x₂ − x₁)² + (y₂ − y₁)²]
Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2)
For the midpoint you add the coordinates; for the length you subtract them
If M is the midpoint of AB, then B = (2 × x of M − x of A, 2 × y of M − y of A)
Useful whole-number triangles: 3, 4, 5; 5, 12, 13; 6, 8, 10; 8, 15, 17

Equations of linear graphs

y = mx + c: m is the gradient, c is the y-intercept
Gradient from two points: m = (y₂ − y₁)/(x₂ − x₁)
To find c, substitute the gradient and one known point into y = mx + c
Another form: y − y₁ = m(x − x₁) for a line with gradient m through (x₁, y₁)
Horizontal line through (a, b): y = b; vertical line through (a, b): x = a
Divide every term when making y the subject: 3y = 2x + 12 becomes y = (2/3)x + 4

Parallel lines

Parallel lines have equal gradients: m₁ = m₂
y = mx + c and y = mx + d are parallel when c and d are different
Rearrange to y = mx + c before comparing gradients
A line parallel to ax + by = c can be written ax + by = k; find k by substituting the point
Line through (x₁, y₁) parallel to a line of gradient m: y − y₁ = m(x − x₁)

Perpendicular lines

Perpendicular lines: gradient of one × gradient of the other = −1 (m₁ × m₂ = −1)
Perpendicular gradient = −1 ÷ (gradient of the given line): 3 → −1/3, −2/5 → 5/2
Rearrange to y = mx + c before reading the gradient: 2x + y = 7 is y = −2x + 7, gradient −2
Gradient between two points = (y₂ − y₁) ÷ (x₂ − x₁)
Midpoint of AB = ((x₁ + x₂)/2, (y₁ + y₂)/2)
Perpendicular bisector of AB: passes through the midpoint of AB, gradient = −1 ÷ (gradient of AB)
A horizontal line (y = k) and a vertical line (x = h) are perpendicular

Geometry

Geometrical terms

Acute: less than 90°. Right: 90°. Obtuse: between 90° and 180°. Reflex: between 180° and 360°
Complementary angles add up to 90°; supplementary angles add up to 180°
Equilateral: 3 equal sides. Isosceles: 2 equal sides. Scalene: no equal sides
Trapezium: one pair of parallel sides. Parallelogram: two pairs. Rhombus: parallelogram with 4 equal sides. Kite: two pairs of equal sides next to each other
Diagonals: equal in a rectangle; cross at 90° in a rhombus and a kite; both in a square
Circle: radius, diameter (chord through the centre), circumference, arc (part of the circumference), sector (between two radii and an arc), segment (between a chord and an arc)
Polygon names: pentagon 5, hexagon 6, heptagon 7, octagon 8, nonagon 9, decagon 10 sides

Geometrical constructions

Triangle from three sides: draw one side; from each end draw an arc with radius equal to another side; join the crossing point to both ends
A triangle can be made only if (sum of the two shorter sides) is greater than the longest side
Perpendicular bisector of AB: equal arcs from A and from B, with radius more than half of AB; join the two crossing points
Angle bisector: arc from the vertex cutting both arms, then equal arcs from those two points; join the vertex to the crossing point
Reflex angle = 360° − the smaller angle at the same point
Nets: cube = 6 squares; cuboid = 6 rectangles; cylinder = 1 rectangle + 2 circles; square-based pyramid = 1 square + 4 triangles
Area of the net = total surface area of the solid

Scale drawings

Scale 1 : n: real length = map length × n; map length = real length ÷ n
100 cm = 1 m and 100 000 cm = 1 km, so 1 : 50 000 means 1 cm represents 0.5 km
Areas use the scale squared: if 1 cm represents 0.5 km, then 1 cm² represents 0.5 × 0.5 = 0.25 km²
Bearing: start at north, turn clockwise, write three figures (7° is written 007°)
North 000°, east 090°, south 180°, west 270°; north-east 045°, south-east 135°, south-west 225°, north-west 315°
Back bearing: add 180° if the bearing is less than 180°, subtract 180° if it is more (110° → 290°, 250° → 070°)

Similarity

Length scale factor k = length on the larger shape ÷ matching length on the smaller shape
Missing length on the larger shape = matching length on the smaller shape × k
Area scale factor = k² (this includes surface area)
Volume scale factor = k³ (this also works for mass and capacity when the material is the same)
Going back: k = √(area ratio) = cube root of (volume ratio)
Line DE parallel to BC in triangle ABC: triangles ADE and ABC are similar, so AD/AB = AE/AC = DE/BC

Symmetry

Regular polygon with n sides: n lines of symmetry and rotational symmetry of order n
Square: 4 lines, order 4. Rectangle: 2 lines, order 2. Rhombus: 2 lines (its diagonals), order 2
Parallelogram: 0 lines, order 2. Kite: 1 line, order 1. Isosceles trapezium: 1 line, order 1
Equilateral triangle: 3 lines, order 3. Isosceles triangle: 1 line, order 1. Scalene triangle: 0 lines, order 1
Cuboid with three different edge lengths: 3 planes of symmetry. Cube: 9 planes of symmetry
Right pyramid on a regular n-sided base: n planes of symmetry (a regular tetrahedron has 6) and rotational symmetry of order n about its axis
Cylinder and cone: an infinite number of planes of symmetry through the axis (a cylinder has one more, halfway along its length)

Angles

Angles at a point = 360°; angles on a straight line = 180°; vertically opposite angles are equal
Triangle = 180°; quadrilateral = 360°; an isosceles triangle has two equal base angles
Exterior angle of a triangle = sum of the two opposite interior angles
Parallel lines: alternate angles are equal (Z shape); corresponding angles are equal (F shape); co-interior angles add up to 180° (C shape)
Sum of the interior angles of a polygon with n sides = (n − 2) × 180°
Sum of the exterior angles of any polygon = 360°; regular polygon: each exterior angle = 360° ÷ n
Interior angle + exterior angle = 180°

Circle theorems I

Angle at the centre = 2 × angle at the circumference on the same arc
Angle in a semicircle = 90° (the angle that a diameter makes at the circumference)
Angles in the same segment are equal (same arc, same side of the chord)
Opposite angles of a cyclic quadrilateral add up to 180°
Exterior angle of a cyclic quadrilateral = the interior opposite angle
Tangent and radius meet at 90° at the point of contact
All radii are equal, so a triangle made by two radii and a chord is isosceles

Circle theorems II

The perpendicular from the centre to a chord bisects the chord; the perpendicular bisector of a chord passes through the centre
radius² = (distance from centre to chord)² + (half the chord)²
Equal chords are the same distance from the centre
Tangents from an outside point P are equal: PA = PB, so triangle PAB is isosceles
In kite OAPB the angles at A and B are 90°, so angle AOB + angle APB = 180°
Tangent, radius and the line OP make a right-angled triangle: OP² = radius² + PA²
Alternate segment theorem: angle between tangent and chord = angle in the alternate segment

Mensuration

Units of measure

Length: 10 mm = 1 cm; 100 cm = 1 m; 1000 m = 1 km
Mass: 1000 mg = 1 g; 1000 g = 1 kg; 1000 kg = 1 tonne
Area: 1 cm² = 100 mm²; 1 m² = 10 000 cm²; 1 km² = 1 000 000 m²
Volume: 1 cm³ = 1000 mm³; 1 m³ = 1 000 000 cm³
Capacity: 1 litre = 1000 ml = 1000 cm³; 1 ml = 1 cm³; 1 m³ = 1000 litres
Speed: km/h ÷ 3.6 = m/s; m/s × 3.6 = km/h (36 km/h = 10 m/s)

Area and perimeter

Rectangle: area = length × width; perimeter = 2 × (length + width)
Triangle: area = 1/2 × base × perpendicular height
Parallelogram: area = base × perpendicular height
Trapezium: area = 1/2 × (sum of the parallel sides) × perpendicular distance between them (A = 1/2 × (a + b) × h)
Rhombus or kite: area = 1/2 × (product of the diagonals)
Square: side = perimeter ÷ 4; area = side²
Path or border: area = area of the outer shape − area of the inner shape

Circles, arcs and sectors

Circumference = 2 × π × radius = π × diameter (C = 2πr = πd)
Area of a circle = π × radius² (A = πr²)
Arc length = (angle ÷ 360) × 2πr
Sector area = (angle ÷ 360) × πr²
Perimeter of a sector = arc length + 2 × radius
Perimeter of a semicircle = πr + 2r (half the circumference plus the diameter)
Angle of a sector = (arc length ÷ circumference) × 360° = (sector area ÷ area of circle) × 360°

Surface area and volume

Cuboid: volume = length × width × height; surface area = 2(lw + lh + wh)
Prism: volume = area of cross-section × length
Cylinder: volume = πr²h; curved surface area = 2πrh; total surface area of a closed cylinder = 2πrh + 2πr²
Pyramid: volume = (1/3) × base area × perpendicular height
Cone: volume = (1/3)πr²h; curved surface area = πrl, where l is the slant height
Sphere: volume = (4/3)πr³; surface area = 4πr²
1 litre = 1000 cm³; 1 m³ = 1 000 000 cm³

Compound shapes and parts of shapes

Joined shapes: total area (or volume) = sum of the parts
Shape with a piece removed: area (or volume) left = whole − piece removed
Semicircle: area = (1/2)πr²; curved edge = πr
Sector with angle θ°: area = (θ/360) × πr²; arc length = (θ/360) × 2πr
Segment: area = area of sector − area of triangle
Ring between two circles with radii R and r: area = πR² − πr²
Hemisphere: volume = (2/3)πr³; curved surface area = 2πr²

Trigonometry

Pythagoras’ theorem

a² + b² = c², where c is the hypotenuse
Hypotenuse: c = √(a² + b²)
Shorter side: a = √(c² − b²)
Test: if the three sides fit a² + b² = c², the triangle is right-angled, with the right angle opposite c
Whole-number sets to know: 3, 4, 5; 5, 12, 13; 8, 15, 17; 7, 24, 25, and their multiples such as 6, 8, 10
Check: the hypotenuse must be longer than each other side, and shorter than their sum

Right-angled triangles

sin x = opposite ÷ hypotenuse
cos x = adjacent ÷ hypotenuse
tan x = opposite ÷ adjacent
To find an angle: x = sin⁻¹(opposite ÷ hypotenuse), and the same with cos⁻¹ and tan⁻¹
Angle of elevation: measured up from the horizontal. Angle of depression: measured down from the horizontal. The angle of depression from A to B equals the angle of elevation from B to A
Bearing: angle measured clockwise from north, three figures, e.g. 050°
The perpendicular from a point to a line is the shortest distance to the line

Non-right-angled triangles

Sine rule for a side: a ÷ sin A = b ÷ sin B = c ÷ sin C
Sine rule for an angle: (sin A) ÷ a = (sin B) ÷ b
Cosine rule for a side: a² = b² + c² − 2bc cos A
Cosine rule for an angle: cos A = (b² + c² − a²) ÷ (2bc)
Area of triangle = (1/2)ab sin C, where C is the angle between sides a and b
Angles of a triangle add up to 180°: find the third angle first if it gives you a matching side and angle
sin(180° − x) = sin x, e.g. sin 150° = sin 30° = 0.5

Pythagoras’ theorem and trigonometry

Two sides known, third side wanted: Pythagoras. An angle known or wanted: sin, cos or tan
Longest diagonal of a cuboid = √(length² + width² + height²)
Angle between a space diagonal and the base of a cuboid: tan x = height ÷ diagonal of the base
Isosceles triangle: the perpendicular from the top vertex cuts the base in half and cuts the top angle in half
Rhombus: the diagonals cut each other in half at right angles
Area of a triangle = (1/2) × base × perpendicular height
Keep the full calculator value in the middle steps and round only at the end

Transformations and vectors

Transformations

Reflection: in the x-axis (x, y) → (x, −y); in the y-axis (x, y) → (−x, y); in y = x (x, y) → (y, x); in y = −x (x, y) → (−y, −x)
Reflection in the line x = a: (x, y) → (2a − x, y). In the line y = b: (x, y) → (x, 2b − y)
Rotation about the origin: 90° anticlockwise (x, y) → (−y, x); 90° clockwise (x, y) → (y, −x); 180° (x, y) → (−x, −y)
Translation by the vector (a, b): (x, y) → (x + a, y + b)
Enlargement, scale factor k: image = centre + k × (vector from centre to point). Lengths × k, area × k²
Scale factor between 0 and 1 makes the image smaller; a negative scale factor puts the image on the opposite side of the centre
The inverse of a rotation is the same angle about the same centre in the opposite direction

Vectors in two dimensions

Translation vector = coordinates of image − coordinates of object
Add: (a, b) + (c, d) = (a + c, b + d)
Subtract: (a, b) − (c, d) = (a − c, b − d)
Multiply by a scalar: k(a, b) = (ka, kb)
Journeys join up: vector AB + vector BC = vector AC
The vector −a has the same length as a but the opposite direction
Left and down are negative: 6 left and 2 down is (−6, −2)

Magnitude of a vector

Magnitude of the vector (x, y) = √(x² + y²)
Vector AB = position vector of B − position vector of A (AB = b − a)
Vector BA = −(vector AB)
Midpoint M of AB: OM = (1/2)(a + b)
P on AB with AP : PB = m : n: OP = a + (m ÷ (m + n)) × (b − a)
Parallel: if vector PQ = k × vector RS, then PQ is parallel to RS and k times as long
Three points A, B, C lie on one straight line if AB and BC are parallel, because they share the point B

Probability

Introduction to probability

P(event) = number of outcomes you want ÷ total number of equally likely outcomes
0 ≤ P(event) ≤ 1: impossible = 0, certain = 1
P(not A) = 1 − P(A), also written P(A′) = 1 − P(A)
The probabilities of all the outcomes add up to 1, when only one of them can happen at a time
Number of items of one kind = P(that kind) × total number of items
Total number of items = number of one kind ÷ P(that kind)
Give a probability as a fraction, decimal or percentage, not as a ratio

Relative and expected frequencies

Relative frequency = number of times the event happens ÷ total number of trials
Relative frequency is an estimate of the probability; more trials give a better estimate
Expected frequency = probability × number of trials
Number of trials = expected frequency ÷ probability
Expected number of times it does not happen = (1 − probability) × number of trials
Fair object: use the theoretical probability. Biased object: use the relative frequency

Probability of combined events

Independent events: P(A and B) = P(A) × P(B)
Mutually exclusive events: P(A or B) = P(A) + P(B)
P(at least one) = 1 − P(none)
Tree diagram: multiply along the branches, add between the routes; each set of branches adds up to 1
Without replacement: the second probability has a total one smaller, e.g. 4/10 then 3/9
Two dice give 6 × 6 = 36 equally likely outcomes; two coins give 4
Venn diagram: number in A or B = number in A + number in B − number in both

Statistics

Classifying statistical data

Qualitative (categorical) data: words or categories, e.g. eye colour.
Quantitative data: numbers. It is either discrete or continuous.
Discrete: counted, only separate values, e.g. number of pets. Continuous: measured, any value in a range, e.g. time.
Shoe size is discrete: only certain sizes exist, even though some are halves.
Groups for continuous data: 10 < x ≤ 20, 20 < x ≤ 30, and so on (no gaps, no overlaps).
Groups for discrete data can be written 0–4, 5–9, 10–14.
For qualitative data the only average that can be found is the mode.

Interpreting statistical data

Total from a frequency table = sum of (value × frequency).
Percentage of the total = (part ÷ total) × 100.
Compare an average (mean or median) to say which set is higher in general.
Compare the spread (range or interquartile range): a smaller spread means more consistent values.
A sample must be large enough and must represent the whole population.
An axis that does not start at 0 makes a small change look large.
Correlation between two things does not show that one causes the other.

Averages and measures of spread

Mean = sum of the values ÷ number of values.
Median = middle value when the values are in order. For n values it is the (n + 1)/2 th value; with two middle values, take the number halfway between them.
Mode = the value that occurs most often. Range = largest value − smallest value.
Frequency table: mean = sum of (value × frequency) ÷ sum of frequencies (Σfx ÷ Σf).
Grouped data: estimate of the mean = sum of (midpoint × frequency) ÷ sum of frequencies.
Modal class = the class with the highest frequency.
Sum of the values = mean × number of values.

Statistical charts and diagrams

Pie chart: sector angle = (frequency ÷ total) × 360°.
Pie chart: frequency = (sector angle ÷ 360°) × total.
Pictogram: number = number of symbols × value of one symbol; count part symbols as fractions.
Stem-and-leaf: leaves are in order and the key gives the meaning, e.g. 2 | 5 means 25.
Frequency polygon: plot each frequency above the midpoint of its class and join the points with straight lines.
Composite (stacked) bar: the value of a section is its top reading − its bottom reading.
Dual bar chart: two bars side by side for each category, used to compare two sets.

Scatter diagrams

Positive correlation: as one quantity increases, the other increases (points slope upwards from left to right).
Negative correlation: as one quantity increases, the other decreases (points slope downwards from left to right).
Zero (no) correlation: the points show no pattern.
The closer the points are to a straight line, the stronger the correlation.
Line of best fit: one straight ruled line through the middle of the points, with points balanced on both sides.
An outlier is a point far from the pattern of the others.
Estimates outside the range of the data are unreliable, because the trend may not continue.

Cumulative frequency diagrams

Cumulative frequency = sum of all the frequencies up to the end of that class.
Plot each point at (upper class boundary, cumulative frequency).
For n values: median at n/2, lower quartile at n/4, upper quartile at 3n/4 on the cumulative frequency axis.
Interquartile range = upper quartile − lower quartile.
The pth percentile is read at (p ÷ 100) × n. The median is the 50th percentile.
Number of values greater than x = total − cumulative frequency at x.
Frequency of one class = its cumulative frequency − the cumulative frequency of the class before.

Histograms

Frequency density = frequency ÷ class width.
Frequency = frequency density × class width = area of the bar.
Class width = upper class boundary − lower class boundary, e.g. 10 < x ≤ 25 has width 15.
The vertical axis is frequency density; the horizontal axis is a continuous scale with no gaps between bars.
Bars of equal height have equal frequency density, not equal frequency.
For part of a class: frequency ≈ frequency density × width of that part.