One of the most common revision mistakes is memorising formulae that are actually given on the exam paper — and failing to memorise the ones that are not. This guide tells you exactly which formulae each board provides, which ones you must know from memory, and how to apply them.
What Is on the GCSE Maths Formula Sheet?
All three major boards — AQA, Edexcel and OCR — provide a formula sheet printed inside each exam paper. The content is similar but not identical. Here is what each board provides:
Formulae Given on All Three Boards
- Area of a trapezium: A = ½(a + b)h
- Volume of a prism: V = area of cross-section × length
- Quadratic formula: x = (−b ± √(b² − 4ac)) / 2a (Higher only)
Formulae Given on Some Boards
- Compound interest: A = P(1 + r/100)ⁿ — provided on some papers; worth memorising anyway
- Reverse percentage method — not a formula, but a method; always show working
Important: The formula sheet does NOT include Pythagoras' theorem, trigonometric ratios (SOH-CAH-TOA), circle area/circumference, or basic area formulas. Students commonly assume these are given — they are not.
Formulae You Must Know From Memory
These are not on the formula sheet. You need to memorise them.
Number and Algebra
Percentage change: $$\text{Percentage change} = \frac{\text{New} - \text{Original}}{\text{Original}} \times 100$$
Compound interest / growth: $$A = P \left(1 + \frac{r}{100}\right)^n$$ Where A = final amount, P = principal, r = percentage rate, n = number of periods.
Compound decay (depreciation): $$A = P \left(1 - \frac{r}{100}\right)^n$$
Nth term of an arithmetic sequence: $$T_n = a + (n-1)d$$ Where a = first term, d = common difference.
Geometry: Lengths and Areas
Area of a rectangle: A = length × width
Area of a triangle: A = ½ × base × height
Area of a parallelogram: A = base × height
Circumference of a circle: C = 2πr = πd
Area of a circle: A = πr²
Arc length (Higher): $$\text{Arc length} = \frac{\theta}{360} \times 2\pi r$$
Area of a sector (Higher): $$\text{Sector area} = \frac{\theta}{360} \times \pi r^2$$
Geometry: Volumes and Surface Areas
Volume of a cuboid: V = length × width × height
Volume of a cylinder: V = πr²h
Volume of a pyramid (Higher): V = ⅓ × base area × height
Volume of a cone (Higher): V = ⅓πr²h
Volume of a sphere (Higher): V = 4/3 πr³
Surface area of a sphere (Higher): SA = 4πr²
Curved surface area of a cone (Higher): SA = πrl (where l = slant height)
Pythagoras and Trigonometry
Pythagoras' theorem: $$a^2 + b^2 = c^2$$ Where c is the hypotenuse (longest side).
SOH-CAH-TOA: $$\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}, \quad \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}, \quad \tan\theta = \frac{\text{opposite}}{\text{adjacent}}$$
Sine rule (Higher): $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$
Cosine rule (Higher): $$a^2 = b^2 + c^2 - 2bc\cos A$$
Area of a triangle using sine (Higher): $$\text{Area} = \frac{1}{2}ab\sin C$$
Exact Trigonometric Values (Higher)
These must be memorised — they appear on non-calculator Higher papers:
| Angle | sin | cos | tan |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | ½ | √3/2 | 1/√3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | ½ | √3 |
| 90° | 1 | 0 | undefined |
Statistics
Mean from a frequency table: $$\bar{x} = \frac{\sum fx}{\sum f}$$ Where f = frequency, x = value (or midpoint for grouped data).
Probability: $$P(\text{event}) = \frac{\text{number of favourable outcomes}}{\text{total number of outcomes}}$$
Combined probability (independent events): $$P(A \text{ and } B) = P(A) \times P(B)$$
$$P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$$
Ratio, Speed and Density
Speed, distance, time: $$\text{Speed} = \frac{\text{Distance}}{\text{Time}}, \quad \text{Distance} = \text{Speed} \times \text{Time}, \quad \text{Time} = \frac{\text{Distance}}{\text{Speed}}$$
Density, mass, volume: $$\text{Density} = \frac{\text{Mass}}{\text{Volume}}, \quad \text{Mass} = \text{Density} \times \text{Volume}$$
Pressure, force, area (Higher): $$\text{Pressure} = \frac{\text{Force}}{\text{Area}}$$
Worked Examples: Applying Key Formulae
Example 1 — Compound Interest
A student invests £2,000 at 3.5% per year compound interest. How much is it worth after 4 years?
$$A = 2000 \times (1 + 3.5/100)^4 = 2000 \times (1.035)^4 = 2000 \times 1.1475 = £2295.05$$
Common mistake: Using simple interest: 2000 × 0.035 × 4 = £280 → total £2280. This earns M0 — wrong method.
Example 2 — Cosine Rule
In a triangle, sides b = 7 cm, c = 5 cm, and angle A = 63°. Find side a.
$$a^2 = 7^2 + 5^2 - 2(7)(5)\cos 63° = 49 + 25 - 70 \times 0.454 = 74 - 31.78 = 42.22$$ $$a = \sqrt{42.22} = 6.50 \text{ cm (to 3 s.f.)}$$
Example 3 — Sector Area
Find the area of a sector with radius 8 cm and angle 135°.
$$\text{Sector area} = \frac{135}{360} \times \pi \times 8^2 = 0.375 \times \pi \times 64 = 75.4 \text{ cm}^2 \text{ (to 3 s.f.)}$$
Example 4 — Volume of a Cone
A cone has radius 4 cm and height 9 cm. Find its volume.
$$V = \frac{1}{3}\pi r^2 h = \frac{1}{3} \times \pi \times 16 \times 9 = 48\pi = 150.8 \text{ cm}^3 \text{ (to 1 d.p.)}$$
Leave in terms of π when asked — "Express your answer in terms of π" means write 48π, not 150.8.
How to Memorise Formulae
Flashcard method: Write the formula name on one side, the formula on the other. Test yourself daily — particularly in the 4 weeks before the exam.
Grouping by topic: Learn related formulae together (all circle formulae in one session, all trigonometry in one session). This builds conceptual connections.
Practice application: Knowing a formula is not the same as being able to use it under time pressure. For each formula you memorise, practise 3–5 questions that require it until applying it feels automatic.
Non-calculator fluency: Several formulae need to be applied without a calculator on Paper 1. Practise exact-value trigonometry and standard form calculations by hand.
Quick-Reference Summary by Topic
| Topic | Key Formulae to Know (not on sheet) |
|---|---|
| Circles | C = 2πr or πd; A = πr² |
| Triangles | Area = ½bh; Pythagoras a²+b²=c²; SOH-CAH-TOA |
| Sequences | Tₙ = a + (n−1)d |
| Percentage change | (New−Original)/Original × 100 |
| Compound growth | P(1 + r/100)ⁿ |
| Speed | S = D/T |
| Density | D = M/V |
| Probability | P = favourable/total |
| Mean (frequency) | Σfx / Σf |
| Higher: Sine rule | a/sin A = b/sin B |
| Higher: Cosine rule | a² = b² + c² − 2bc cos A |
| Higher: Sector area | (θ/360) × πr² |
| Higher: Cone volume | ⅓πr²h |
| Higher: Sphere volume | 4/3 πr³ |
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