Formula Study Method
For Mathematics, marks are earned by method as well as final answers. A learner should copy each formula into a formula book, add one worked example, then practise questions until the method feels automatic. Geometry theorems must be stated accurately and then applied to the diagram with a reason.
Grade 8 mathematics
Grade 8 Formulas and Theorems
Number work, perimeter, area, basic geometry and first algebraic rules.
| Formula | Notation | How to use it |
|---|---|---|
| Rectangle perimeter | P = 2(l + w) | Add length and width, then double the result. |
| Square perimeter | P = 4s | Multiply the side length by four. |
| Rectangle area | A = l × w | Multiply length by width. |
| Square area | A = s2 | Square the side length. |
| Triangle area | A = 1/2 × b × h | Use base and perpendicular height. |
| Parallelogram area | A = b x h | Use base and perpendicular height. |
| Circle circumference | C = 2πr or C = πd | Use radius or diameter. |
| Circle area | A = πr2 | Square the radius, then multiply by π. |
| Percentage | percentage = part / whole × 100 | Use for discounts, increases and comparison. |
| Speed | speed = distance / time | Use matching units before calculating. |
Theorems and facts to know
Angles on a straight line
Angles that form a straight line add to 180 degrees.
Angles around a point
Angles around one point add to 360 degrees.
Vertically opposite angles
Vertically opposite angles are equal.
Triangle angle sum
The three interior angles of a triangle add to 180 degrees.
Parallel-line angle facts
Corresponding angles are equal, alternate angles are equal, and co-interior angles add to 180 degrees.
Grade 9 mathematics
Grade 9 Formulas and Theorems
Exponents, algebra, factorising, pythagoras, similarity and geometry proofs.
| Formula | Notation | How to use it |
|---|---|---|
| Exponent product | am x an = am+n | Use when multiplying powers with the same base. |
| Exponent quotient | am / an = am-n | Use when dividing powers with the same base. |
| Power of a power | (am)n = amn | Multiply the exponents. |
| Zero exponent | a0 = 1 | True when a is not zero. |
| Difference of squares | a2 - b2 = (a - b)(a + b) | Use for fast factorising. |
| Pythagoras | c2 = a2 + b2 | Use only in a right-angled triangle; c is the hypotenuse. |
| Mean | mean = sum of values / number of values | Use for average. |
| Simple interest | I = P × r × t | Use rate as a decimal unless the question says otherwise. |
Theorems and facts to know
Exterior angle of a triangle
An exterior angle of a triangle equals the sum of the two opposite interior angles.
Congruent triangles
Triangles can be congruent by SSS, SAS, ASA or RHS.
Similar triangles
Similar triangles have equal corresponding angles and proportional corresponding sides.
Pythagoras theorem
In a right-angled triangle, the square on the hypotenuse equals the sum of the squares on the other two sides.
Midpoint idea
A midpoint divides a line segment into two equal parts.
Grade 10 mathematics
Grade 10 Formulas and Theorems
Functions, analytical geometry, trigonometry, finance, measurement and algebra.
| Formula | Notation | How to use it |
|---|---|---|
| Linear gradient | m = (y2 - y1) / (x2 - x1) | Use two points on a straight line. |
| Midpoint | M = ((x1 + x2) / 2 ; (y1 + y2) / 2) | Find the middle of a line segment. |
| Distance | d = sqrt((x2 - x1)2 + (y2 - y1)2) | Find the length between two coordinate points. |
| Straight line | y = mx + c | m is the gradient and c is the y-intercept. |
| Trigonometric ratios | sin θ = opposite / hypotenuse; cos θ = adjacent / hypotenuse; tan θ = opposite / adjacent | Use in right-angled triangles. |
| Trig identity | tan θ = sin θ / cos θ | Use to simplify trigonometric expressions. |
| Simple interest | A = P(1 + in) | Use for non-compounding growth. |
| Compound interest | A = P(1 + i)n | Use when interest compounds each period. |
| Cylinder volume | V = πr2h | Use radius and height. |
| Sphere volume | V = 4/3πr3 | Use radius cubed. |
Theorems and facts to know
Gradient of parallel lines
Parallel lines have equal gradients.
Gradient of perpendicular lines
Perpendicular line gradients multiply to -1.
Trigonometry in right triangles
The chosen ratio must match the sides given and the side required.
Triangle proportion
Similar triangles have equal angles and proportional sides.
Quadrilateral facts
Use the defining properties of parallelograms, rectangles, rhombi, squares and trapezia to prove unknown sides or angles.
Grade 11 mathematics
Grade 11 Formulas and Theorems
Quadratic functions, euclidean geometry, trigonometric identities, sequences and finance.
| Formula | Notation | How to use it |
|---|---|---|
| Quadratic function | y = ax2 + bx + c | Use to identify shape, intercepts and turning point. |
| Turning point axis | x = -b / 2a | Use for the axis of symmetry of a quadratic. |
| Quadratic formula | x = (-b ± sqrt(b2 - 4ac)) / 2a | Use when factorising is difficult or impossible. |
| Arithmetic sequence | Tn = a + (n - 1)d | Use common difference d. |
| Geometric sequence | Tn = arn-1 | Use common ratio r. |
| Sine rule | a / sin A = b / sin B = c / sin C | Use in non-right triangles when a side-angle opposite pair is known. |
| Cosine rule | a2 = b2 + c2 - 2bc cos A | Use in non-right triangles with SAS or SSS information. |
| Area rule | area = 1/2 ab sin C | Use two sides and the included angle. |
| Compound decay | A = P(1 - i)n | Use for depreciation or decreasing value. |
Theorems and facts to know
Line from centre to chord
The perpendicular from the centre of a circle to a chord bisects the chord.
Equal chords
Equal chords subtend equal angles at the centre.
Angle at centre
The angle at the centre is twice the angle at the circumference standing on the same arc.
Angles in the same segment
Angles in the same segment of a circle are equal.
Opposite angles of cyclic quadrilateral
Opposite angles of a cyclic quadrilateral are supplementary.
Exterior angle of cyclic quadrilateral
The exterior angle of a cyclic quadrilateral equals the interior opposite angle.
Grade 12 mathematics
Grade 12 Formulas and Theorems
Matric algebra, calculus, finance, sequences, analytical geometry, trigonometry and euclidean geometry.
| Formula | Notation | How to use it |
|---|---|---|
| Arithmetic series | Sn = n/2[2a + (n - 1)d] | Use when adding arithmetic terms. |
| Geometric series | Sn = a(rn - 1) / (r - 1) | Use when adding geometric terms and r is not 1. |
| Infinite geometric series | S∞ = a / (1 - r) | Use only when -1 < r < 1. |
| Derivative from first principles | f'(x) = limh→0 [f(x + h) - f(x)] / h | Use to prove or derive a derivative. |
| Power rule | d/dx(xn) = nxn-1 | Use for polynomial differentiation. |
| Cubic function | f(x) = ax3 + bx2 + cx + d | Use with intercepts, turning points and shape. |
| Compound interest | A = P(1 + i)n | Use effective periods carefully. |
| Present value | P = A(1 + i)-n | Discount a future amount back to today. |
| Trigonometric identity | sin2θ + cos2θ = 1 | Use for simplification and proofs. |
| Double angle | sin 2θ = 2 sin θ cos θ; cos 2θ = cos2θ - sin2θ | Use in trig equations and identities. |
Theorems and facts to know
Tangent-chord theorem
The angle between a tangent and a chord equals the angle in the alternate segment.
Circle tangent radius
A tangent to a circle is perpendicular to the radius at the point of contact.
Cyclic quadrilateral
Opposite angles of a cyclic quadrilateral add to 180 degrees.
Proportionality theorem
A line parallel to one side of a triangle divides the other two sides proportionally.
Similarity proof
To prove similarity, show equal corresponding angles or proportional corresponding sides.
Stationary points
Turning points occur where f'(x) = 0, then the shape is confirmed using signs, a table or the second derivative where appropriate.
Exam warning
Common Formula Mistakes That Lose Marks
Using the wrong triangle side
In trigonometry, label opposite, adjacent and hypotenuse before choosing sin, cos or tan.
Forgetting squared units
Area answers use square units and volume answers use cubic units.
Mixing up radius and diameter
If the question gives diameter, divide by 2 before using formulas with radius.
Skipping reasons in geometry
Geometry answers need statements and reasons, not only numbers on the diagram.
Wrong finance period
Match the interest rate period with the number of compounding periods.
Calculator mode errors
Most school trigonometry questions use degrees. Check calculator mode before answering.