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IGCSE Mathematics 0580 · Formula sheet · ~10 min read

Maths Formula & Method Sheet

Every formula you must be able to write down from memory, drawn where a picture helps, plus the first move for the most common question types. One sheet for the night before.

NumberAlgebraGeometryMensurationTrigonometryStatisticsProbability
RSby Riyaz Sir
mentorriyaz.in
No tag = Core (everyone)EXTENDED Extended onlyhighlight = comes up again and againHANDY worth memorisingTest yourself at the end

In 30 seconds

  • Learn these cold: percentage change, indices, gradient, angle rules, area and volume, and SOH CAH TOA. Each one saves you thinking time on every paper.
  • Some formulas are printed on your paper (such as the curved areas and volumes of a cone, sphere and pyramid, and for Extended the quadratic formula and the sine and cosine rules). You still need to rearrange and use them correctly.
  • Method marks: write the formula, substitute, then calculate. Keep full calculator values and round only at the end (3 significant figures, angles to 1 decimal place).

1Number

Percentages, ratio and speed

percentage change = changeoriginal × 100
always divide by the ORIGINAL value
original = new amountmultiplier
reverse percentage. After a 20% rise the multiplier is 1.2
compound interest: final = P(1 + r100)n
P = start amount, r = % rate, n = years
simple interest = P × r × n100
the same amount is added every year
speed = distancetime
also density = massvolume
  • Multipliers: a 15% increase is × 1.15. A 15% decrease is × 0.85.
  • Ratio: add the parts to get the total. Share 40 in the ratio 3 : 5 → 40 ÷ 8 = 5, so 15 and 25.
  • Average speed = total distance ÷ total time. It is never the mean of two speeds.

Indices, surds and bounds

am × an = am+n
add the powers when you multiply
am ÷ an = am−n
subtract the powers when you divide
(am)n = amn
multiply the powers
a−n = 1an
a negative power means the reciprocal
a1/n = ⁿ√a
a fractional power is a root. Also a0 = 1
  • Standard form is a × 10n with 1 ≤ a < 10. For example 45 000 = 4.5 × 104 and 0.0006 = 6 × 10−4.
  • Bounds: go half a unit either side. A length of 8.4 cm (1 d.p.) has 8.35 ≤ length < 8.45.
  • Surds EXT: √12 = 2√3, and to rationalise 1√3 multiply top and bottom by √3 to get √33.

2Algebra

Expanding, factorising and solving

(a + b)² = a² + 2ab + b²
and (a − b)² = a² − 2ab + b². Never a² + b²
a² − b² = (a + b)(a − b)
difference of two squares
x = −b ± √(b² − 4ac)2a
for ax² + bx + c = 0 EXT
x² + bx = (x + b2)² − (b2)²
completing the square EXT
  • The discriminant b² − 4ac tells you the roots EXT: positive means 2 real roots, zero means 1 repeated root, negative means no real roots.
  • Inequalities: flip the sign when you multiply or divide by a negative number.

Sequences and straight lines

nth term = a + (n − 1)d
arithmetic sequence, first term a, common difference d
gradient = y₂ − y₁x₂ − x₁
and the line is y = mx + c
midpoint = (x₁ + x₂2, y₁ + y₂2)
the mean of the coordinates
length = √((x₂ − x₁)² + (y₂ − y₁)²)
Pythagoras between two points
  • Parallel lines have the same gradient. Perpendicular lines have gradients that multiply to −1.

Variation and calculusEXTENDED

  • Direct proportion: y = kx, or y = kx². Inverse proportion: y = k/x, or y = k/x². Find k from one pair of values first.
d/dx (axⁿ) = naxⁿ⁻¹
multiply by the power, then reduce the power by 1
  • Turning points are where dy/dx = 0. The gradient of a curve at a point is dy/dx there.
  • Functions: f(g(x)) means do g first. For f⁻¹(x), swap x and y and rearrange.

3Geometry and mensuration

Angles and polygons

60°60°Corresponding (F)angles are equalsame position60°60°Alternate (Z)angles are equalinside, opposite sides120°60°Co-interior (C)angles add up to 180°inside, same side
Parallel lines give three pairs of equal or supplementary angles. Name the rule in your answer.
  • Straight line 180° · around a point 360° · vertically opposite angles are equal · triangle 180° · quadrilateral 360°.
  • Exterior angle of a triangle = the sum of the two opposite interior angles.
interior angles add up to (n − 2) × 180°
n = number of sides
exterior angle of a regular polygon = 360°n
interior angle = 180° − exterior angle
  • Congruent triangles: SSS, SAS, ASA or AAS, and RHS.
  • Similar shapes: if the length scale factor is k, areas scale by k² and volumes by k³.

Circle theoremsEXTENDED

  • Angle in a semicircle = 90°. Angle at the centre = 2 × the angle at the circumference.
  • Angles in the same segment are equal. Opposite angles of a cyclic quadrilateral add to 180°.
  • A tangent is at 90° to the radius. Two tangents from one point are equal in length.
  • Alternate segment theorem: the angle between a tangent and a chord = the angle in the alternate segment.

Area and perimeter

RectanglelwA = l × wP = 2(l + w)TrianglebhA = ½ × b × hh is at right angles to bParallelogrambhA = b × hh is the vertical heightTrapeziumabhA = ½(a + b) × ha and b are parallelCirclerA = πr²C = 2πr (or πd)Sectorθrarc = θ/360 × 2πrarea = θ/360 × πr²
Area is always in squared units. Sector and arc use the fraction θ/360 of the whole circle.
  • Compound shapes: split into rectangles, triangles and circles, or subtract a hole from a larger shape.

Volume and surface area

Prism and cuboidlhwV = area of end × lengthcuboid: V = l × w × hCylinderrhV = πr²hcurved area = 2πrh+ 2 circles = 2πr²ConehrlV = ⅓πr²hcurved area = πrll is the slant heightSphererV = 4/3 × πr³A = 4πr²PyramidhV = ⅓ × base area × hh is the vertical heightUnits to remember1 m = 100 cm = 1000 mm1 m² = 10 000 cm²1 m³ = 1 000 000 cm³1 cm³ = 1 ml1000 cm³ = 1 litre1 litre = 1000 ml
Volume is in cubed units. Prisms, cylinders and cuboids all use area of cross-section × length.
  • Total surface area: cylinder = 2πrh + 2πr². Cone = πrl + πr². Add or leave out the circle(s) carefully.
  • Slant height of a cone: l² = r² + h² by Pythagoras.
  • Same units first: convert everything to cm (or m) before you calculate.

4Trigonometry

Right-angled and non-right-angled triangles

Right-angled trianglesAny triangleθhypotenuseoppositeadjacentsin θ = opp ÷ hypcos θ = adj ÷ hyptan θ = opp ÷ adjSOH CAH TOAABCabcsine rule: a/sin A = b/sin B = c/sin Ccosine rule: a² = b² + c² − 2bc cos Aarea = ½ ab sin Csides a, b, c are opposite A, B, C
Label the hypotenuse first, then the opposite and adjacent sides relative to the angle you are using.
angle0°30°45°60°90°
sin01/2√2/2√3/21
cos1√3/2√2/21/20
tan01/√31√3undefined
  • Pythagoras: a² + b² = c², where c is the hypotenuse. Subtract to find a shorter side.
  • Bearings are measured clockwise from North, using three figures (for example 045° or 270°).
  • Angle of elevation is measured up from the horizontal. Angle of depression is measured down from it.
  • Graphs: sin x and cos x repeat every 360° and lie between −1 and 1. tan x repeats every 180°. EXT

5Statistics and probability

Statistics

mean = Σ(f × x)Σf
for grouped data use the midpoints, and the answer is an estimate
frequency density = frequencyclass width
in a histogram the area of a bar is the frequency EXT
pie chart angle = frequencytotal × 360°
and the angles add to 360°
  • Quartiles EXT: Q1 is at position (n + 1) ÷ 4, the median at (n + 1) ÷ 2 and Q3 at 3(n + 1) ÷ 4. IQR = Q3 − Q1.
  • Cumulative frequency EXT: plot at the upper class boundary and read at ¼n, ½n and ¾n.
  • Scatter graphs: draw a straight line of best fit through the middle of the points. Do not predict outside the data.

Probability

P(event) = number of favourable outcomestotal number of outcomes
outcomes must be equally likely
  • P(not A) = 1 − P(A). Probabilities are between 0 and 1, never a ratio.
  • Independent events: P(A and B) = P(A) × P(B). Mutually exclusive: P(A or B) = P(A) + P(B).
  • Expected frequency = probability × number of trials.
  • Without replacement EXT: the denominators drop by 1 on the second set of branches.

6Quick methods

Question → first move

If you seeDo this first
show thatwrite every step. The last line must equal the value in the question
simplify fullyfactorise top and bottom, then cancel
boundshalf the rounding unit either side
reverse percentagedivide by the multiplier
two sides and the angle between themcosine rule
a side and the angle opposite itsine rule
find an angle in a right-angled trianglesin⁻¹, cos⁻¹ or tan⁻¹
similar shapes, area or volumesquare or cube the length scale factor
turning point EXTdifferentiate and set dy/dx = 0
histogram EXTthe area of a bar is the frequency
cumulative frequency EXTread at ¼n, ½n and ¾n
no calculatorwrite each step. Use exact values such as π and surds
Exam tipBeing given a formula is not enough. Practise rearranging, for example r from V = 4/3 πr³: r = ∛(3V ÷ 4π).

7Test yourself

Test yourself — cover the answers first

Quick check

  1. Write the formula for percentage change.
  2. A price rises by 20% to $60. What was the original price?
  3. Write am × an as a single power.
  4. What is a−2 as a fraction?
  5. Give the gradient formula for two points.
  6. What is the sum of the interior angles of a hexagon?
  7. Write the area of a trapezium.
  8. Write the volume of a cone and a sphere.
  9. Write sin, cos and tan as ratios.
  10. EXT Write the cosine rule for side a.
  11. EXT Write the quadratic formula.
  12. EXT Differentiate 5x³.
Show answers
  1. change ÷ original × 100
  2. $50 (60 ÷ 1.2)
  3. am+n
  4. 1 ÷ a²
  5. (y₂ − y₁) ÷ (x₂ − x₁)
  6. 720°
  7. ½(a + b) × h
  8. ⅓πr²h and 4/3 πr³
  9. opp ÷ hyp, adj ÷ hyp, opp ÷ adj
  10. a² = b² + c² − 2bc cos A
  11. x = (−b ± √(b² − 4ac)) ÷ 2a
  12. 15x²
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