76 formulas · JSS 1 – SS 3
Formula Library
Every formula you need for class, BECE and WASSCE, with what each letter means, a worked example and a picture to show why it works.
76 formulas found
Number & Sets(7)
- Laws of indicesam × an = am + nSimplify powers of the same base without working out every multiplication.
- Negative and fractional indicesa−n = 1anTurn negative and fractional powers into fractions and roots.
- Definition of a logarithmIf ax = N, then loga N = xSwitch between index form and log form: a logarithm is just the power.
- Laws of logarithmslog (xy) = log x + log yCombine or split logarithms, just like the laws of indices.
- Rules of surds√(ab) = √a × √bSimplify square roots and clear roots from the bottom of a fraction (rationalising).
- Converting a number base to base tenvalue = Σ (digit × baseplace)Find the base-ten value of a number written in base 2, 5, 8 or any base.
- Number of elements in a unionn(A ∪ B) = n(A) + n(B) − n(A ∩ B)Count the members of either set without counting the overlap twice.
Algebra(12)
- Perfect squares(a + b)2 = a2 + 2ab + b2Expand a bracket squared in one step.
- Difference of two squaresa2 − b2 = (a + b)(a − b)Factorise one square minus another, or do quick mental sums.
- Quadratic formulax = −b ± √(b2 − 4ac)2aSolve any quadratic equation, even one that won't factorise.
- Discriminant (nature of roots)D = b2 − 4acTell what kind of roots a quadratic has before solving it.
- Sum and product of rootsα + β = −baFind the sum and product of the roots without solving, or build an equation from its roots.
- nth term of an arithmetic progression (AP)Tn = a + (n − 1)dFind any term of a sequence that goes up (or down) by the same amount each time.
- Sum of an arithmetic progressionSn = n2(2a + (n − 1)d)Add up the first n terms of an AP quickly.
- nth term of a geometric progression (GP)Tn = arn − 1Find any term of a sequence that is multiplied by the same number each time.
- Sum of a geometric progressionSn = a(rn − 1)r − 1 (r > 1)Add up the first n terms of a GP.
- Sum to infinity of a GPS∞ = a1 − r (only when −1 < r < 1)Find what an endless GP adds up to when its terms keep shrinking.
- Direct variationy ∝ x means y = kxWhen one quantity doubles, the other doubles too.
- Inverse variationy ∝ 1x means y = kxWhen one quantity doubles, the other halves (e.g. more workers, fewer days).
Geometry(10)
- Angles in a triangleA + B + C = 180°Find a missing angle in any triangle.
- Interior angles of a polygonSum of interior angles = (n − 2) × 180°Find the total, or each, interior angle of a shape with n sides.
- Exterior angles of a polygonSum of exterior angles = 360°Find the number of sides of a regular polygon from one angle.
- Pythagoras' theoremc2 = a2 + b2Find a missing side of a right-angled triangle.
- Gradient of a straight linem = y2 − y1x2 − x1Measure how steep a line is from two points on it.
- Equation of a straight liney = mx + cWrite the equation of a line from its gradient and a point on it.
- Distance between two pointsd = √((x2 − x1)2 + (y2 − y1)2)Find the length of a line joining two points (it is Pythagoras on a grid).
- Midpoint of a lineM = (x1 + x22, y1 + y22)Find the point exactly halfway between two points.
- Angle at the centre of a circleAngle at centre = 2 × angle at circumferenceFind angles inside a circle standing on the same arc.
- Cyclic quadrilateralOpposite angles add up to 180°Find angles in a four-sided shape whose corners all touch a circle.
Mensuration(14)
- Rectangle: area and perimeterArea = l × bMeasure the space inside a rectangle and the distance round it.
- Area of a triangleArea = 12 × base × heightFind the area of any triangle.
- Area of a trapezium and parallelogramTrapezium: Area = 12(a + b)hFind the area of four-sided shapes with parallel sides.
- Circumference of a circleC = 2πr = πdFind the distance round a circle.
- Area of a circleA = πr2Find the space inside a circle.
- Length of an arcL = θ360° × 2πrFind the length of part of a circle's edge.
- Area of a sectorA = θ360° × πr2Find the area of a 'slice' of a circle.
- Volume of a cube and cuboidCuboid: V = l × b × hFind how much space a box takes up.
- Surface area of a cuboidSurface area = 2(lb + bh + lh)Find the total area of all six faces of a box (e.g. paper to wrap it).
- Volume of a prism and cylinderPrism: V = area of cross-section × lengthFind the volume of any solid with the same shape all the way through.
- Surface area of a cylinderCurved surface = 2πrhFind the area of the label on a tin, or of the whole tin.
- Volume of a cone and pyramidPyramid: V = 13 × base area × hFind the volume of pointed solids: exactly one third of the matching prism.
- Surface area of a coneCurved surface = πrlFind the area of the sloping side of a cone (and its base).
- Sphere: volume and surface areaV = 43πr3Measure a ball.
Trigonometry(7)
- Trigonometric ratios (SOH CAH TOA)sin θ = oppositehypotenuseFind sides and angles in a right-angled triangle.
- Special angles (30°, 45°, 60°)sin 30° = 12, cos 30° = √32, tan 30° = 1√3Exact values you can use without tables or a calculator.
- Basic trigonometric identitiessin2 θ + cos2 θ = 1Find one ratio when you know another.
- Sine ruleasin A = bsin B = csin CSolve a triangle that is not right-angled when you know a side and the angle opposite it.
- Cosine rulea2 = b2 + c2 − 2bc cos AFind the third side from two sides and the angle between them, or an angle from three sides.
- Angles of elevation and depressionheight = distance × tan θFind the height of a tree, pole or building from how far away you are and the angle you look up.
- Bearings and back bearingsBack bearing = bearing + 180° (if the bearing is less than 180°)Find the bearing for the return journey.
Statistics(7)
- Mean (average)x̄ = ΣxnShare the total equally: the everyday 'average'.
- Mean from a frequency tablex̄ = ΣfxΣfFind the mean when the data comes in a table with frequencies.
- MedianMedian position = n + 12th value (data in order)Find the middle value, which isn't pulled up or down by extreme values.
- Range and interquartile rangeRange = highest − lowestMeasure how spread out the data is.
- Mean deviationM.D. = Σ∣x − x̄∣nThe average distance of the values from the mean.
- Variance and standard deviationVariance σ2 = Σ(x − x̄)2nA key measure of how spread out data is, often examined in WASSCE.
- Pie chart angleAngle = frequencytotal frequency × 360°Work out the size of each slice when drawing a pie chart.
Probability(5)
- Probability of an eventP(E) = number of favourable outcomestotal number of possible outcomesMeasure how likely something is, from 0 (impossible) to 1 (certain).
- Probability of an event not happeningP(not E) = 1 − P(E)Find the chance that something does not happen.
- Addition rule (OR)Mutually exclusive: P(A or B) = P(A) + P(B)Find the chance of one event OR another.
- Multiplication rule (AND)Independent events: P(A and B) = P(A) × P(B)Find the chance of two things both happening when one doesn't affect the other.
- Expected frequencyExpected frequency = P(E) × number of trialsPredict how many times something should happen in many tries.
Financial Mathematics(8)
- Percentage increase and decreasePercentage change = new − oldold × 100%Compare prices, salaries or scores before and after a change.
- Profit and loss per centProfit = S.P. − C.P.Measure how well a trader did.
- DiscountDiscount = rate100 × marked priceFind the price after a sale reduction.
- CommissionCommission = rate100 × value of salesWork out what an agent earns for selling goods.
- Sharing in a given ratioShare = your part of the ratiosum of the ratio × totalDivide money or items fairly in a given ratio.
- Simple interestI = PRT100Find the interest on a loan or savings when interest is only paid on the starting sum.
- Compound interestA = P(1 + r100)nFind savings when interest is added each year and then earns interest too.
- DepreciationV = P(1 − r100)nFind the value of something that loses a fixed percentage of its value each year, like a car.
Calculus, Matrices & Vectors(6)
- Differentiation: power ruleIf y = axn, then dydx = naxn − 1Find the gradient function of a curve.
- Stationary (turning) pointsAt a turning point, dydx = 0Find the maximum or minimum of a curve.
- Integration: power rule∫ xn dx = xn + 1n + 1 + C (n ≠ −1)Reverse differentiation, and find the area under a curve.
- Determinant of a 2 × 2 matrixFor A = abcd, |A| = ad − bcFind the determinant, needed for the inverse and for solving equations with matrices.
- Inverse of a 2 × 2 matrixA−1 = 1ad − bc d−b−caUndo a matrix, e.g. to solve simultaneous equations.
- Magnitude of a vectorFor v = xy, |v| = √(x2 + y2)Find the length of a vector.