Every Grade 10 maths formula: exponents, factorising, number patterns, functions, finance, probability, trigonometry, analytical geometry and measurement.
This sheet lists every formula and identity in the CAPS Grade 10 syllabus, grouped by Paper 1 and Paper 2 topics. Some, like the finance and analytical geometry formulas, are usually printed on the exam's information sheet; others, like the exponent laws and special-angle values, you must know by heart.
Knowing what each symbol means matters as much as the formula itself, so each row has a short note on when and how to use it.
| Formula | Expression | Note |
|---|---|---|
| Exponent laws | aᵐ·aⁿ = aᵐ⁺ⁿ; aᵐ ÷ aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ | Bases must be the same. |
| Zero and negative exponents | a⁰ = 1; a⁻ⁿ = 1/aⁿ | a ≠ 0. |
| Difference of two squares | a² − b² = (a − b)(a + b) | Take out a common factor first. |
| Sum of two cubes | a³ + b³ = (a + b)(a² − ab + b²) | Last term in the second bracket is positive. |
| Difference of two cubes | a³ − b³ = (a − b)(a² + ab + b²) | The bracket keeps the sign of the question. |
| Formula | Expression | Note |
|---|---|---|
| Linear pattern | Tₙ = bn + c | b is the constant difference. |
| Simple interest | A = P(1 + in) | Hire purchase uses simple interest on the balance. |
| Compound interest | A = P(1 + i)ⁿ | Also used for inflation and population growth. |
| Addition rule | P(A or B) = P(A) + P(B) − P(A and B) | For mutually exclusive events P(A and B) = 0. |
| Complementary events | P(not A) = 1 − P(A) | The event and its complement cover every outcome. |
| Formula | Expression | Note |
|---|---|---|
| Straight line | y = ax + q | a is the gradient; q is the y-intercept. |
| Parabola | y = ax² + q | Turning point (0; q); a > 0 opens up. |
| Hyperbola | y = a/x + q | Asymptotes x = 0 and y = q. |
| Exponential | y = a·bˣ + q (b > 0, b ≠ 1) | Horizontal asymptote y = q. |
| Trig graphs | y = a sin θ + q; y = a cos θ + q; y = a tan θ + q | Amplitude |a| for sin and cos. |
| Formula | Expression | Note |
|---|---|---|
| Ratios | sin θ = o/h; cos θ = a/h; tan θ = o/a | o, a and h relative to the angle θ. |
| Reciprocals | cosec θ = 1/sin θ; sec θ = 1/cos θ; cot θ = 1/tan θ | Defined where the denominator isn't zero. |
| Cartesian plane | sin θ = y/r; cos θ = x/r; tan θ = y/x | r = √(x² + y²) is always positive. |
| Special angles | sin 30° = ½; cos 60° = ½; tan 45° = 1; sin 60° = √3/2 | Learn these from the special triangles. |
| Formula | Expression | Note |
|---|---|---|
| Distance | d = √[(x₂ − x₁)² + (y₂ − y₁)²] | Length of a line segment. |
| Midpoint | M = ((x₁ + x₂)/2; (y₁ + y₂)/2) | Average of the coordinates. |
| Gradient | m = (y₂ − y₁)/(x₂ − x₁) | Parallel: m₁ = m₂. Perpendicular: m₁ × m₂ = −1. |
| Pyramid and cone volume | V = ⅓ × area of base × h | h is the perpendicular height. |
| Sphere | V = (4/3)πr³; SA = 4πr² | Halve both for a hemisphere's curved part. |
Most Grade 10 papers include an information sheet with the finance, analytical geometry and measurement formulas. You still need the exponent laws, factorising patterns and special-angle values by heart.
The first is simple interest (interest on the original amount only). The second is compound interest (interest on interest), which grows faster.
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