Mastering the required formulas is a crucial step towards success in your GCSE Maths Higher Tier exam. While you will be given a formula sheet in the exam, knowing how and when to use each one is the real key. This guide walks you through every essential formula, categorised by topic, with tips on application and common pitfalls.
The Formula Sheet: Your Best Friend
First, familiarise yourself with the layout of the official formula sheet provided by your exam board (AQA, Edexcel, OCR). Knowing where to find a formula quickly saves valuable time. This guide references the formulas you will find there, plus a few extra derivations you need to know by heart.
1. Algebra: The Foundation
Quadratic Formula
For solving equations of the form ax² + bx + c = 0:
x = [-b ± √b² - 4ac] / 2a
How to use it: Identify a, b, and c from your equation. Substitute them carefully into the formula, remembering the ± symbol gives you two solutions. The expression under the square root, b² - 4ac, is called the discriminant. It tells you about the roots:
> 0: Two distinct real roots.
= 0: One repeated real root.
- < 0: No real roots.
Completing the Square
For rewriting a quadratic in the form a(x + p)² + q:
For x² + bx + c, it becomes (x + b2)² - (b2)² + c.
This is vital for finding the turning point of a quadratic graph (vertex) at (-p, q).
2. Geometry and Measures
Area and Volume
You must know these. A common mistake is confusing length, area, and volume units (e.g., cm vs cm² vs cm³).
Area of a trapezium: A = 12 (a + b) h (where a and b are the parallel sides).
Circumference of a circle: C = πd or C = 2πr.
Area of a circle: A = πr².
- Pythagoras' Theorem: (For right-angled triangles)
a² + b² = c², where c is the hypotenuse.
flowchart TD
A["Right-Angled Triangle?"] -->|Yes| B["Identify Hypotenuse, longest side, opposite right angle"]
A -->|No| C["Pythagoras does NOT apply"]
B --> D["Square the two shorter sides,, add them, square root for hypotenuse, a² + b² = c²"]
B --> E["Square the hypotenuse and one short side,, subtract, square root for other side, c² - a² = b²"]
Volume Formulas
Prism: Volume = Area of cross-section × length
Cylinder: V = πr²h
Sphere: V = 43 πr³
Cone: V = 13 πr²h
- Pyramid:
V = 13 × area of base × height
3. Trigonometry
Right-Angled Trigonometry (SOH CAH TOA)
This mnemonic is essential for finding missing sides and angles.
Sin θ = Opposite / Hypotenuse
Cos θ = Adjacent / Hypotenuse
- Tan θ = Opposite / Adjacent
Sine Rule and Cosine Rule
For non-right-angled triangles.
- Sine Rule:
asin A = bsin B = csin C
Use it when you have a pair: an angle and its opposite side.
- Cosine Rule:
a² = b² + c² - 2bc cos A
Use it to find a side given two sides and the included angle, or to find an angle given all three sides.
4. Compound Measures
These are formulas that link different quantities.
Speed, distance, time: Speed = Distance / Time
Density, mass, volume: Density = Mass / Volume
- Pressure, force, area:
Pressure = Force / Area
Remember to use consistent units (e.g., metres and seconds for speed in ms).
5. Probability
Conditional Probability
For tree diagrams and two-way tables, remember:
P(A and B) = P(A) × P(B given A)
This is the multiplication rule. For independent events, it simplifies to P(A) × P(B).
6. Statistics (Higher Tier)
Histograms
No single formula, but a key concept: Frequency Density = Frequency / Class Width. The area of each bar represents the frequency.
quadrantChart
title "Histogram Bar Area = Frequency"
x-axis "Class Width" --> "Wider class = shorter bar? Not necessarily!"
y-axis "Frequency Density" --> "Height = Freq Density"
"Bar Area": 1
7. Key Formulas NOT on the Sheet
You must memorise these:
Percentage change: Percentage Change = (Change / Original) × 100
Gradient of a line: m = (y₂ - y₁) / (x₂ - x₁)
Equation of a line: y = mx + c (gradient-intercept form).
Compound interest / growth & decay: Final = Original × (Multiplier)(number of periods).
- Quadratic sequences: The nth term of a quadratic sequence is of the form an² + bn + c. Find 2a from the second difference, then work out b and c.
Step-by-Step Strategy for Using Formulas in Exams
Identify the topic. Is it geometry, algebra, trig? This points you to the right formula family.
Write down the formula. Get it from your memory or the formula sheet and write it clearly on your paper.
Substitute carefully. Replace each letter in the formula with the corresponding number from the question. Use brackets to avoid errors.
Calculate step-by-step. Don't try to do it all in one go on your calculator.
- Check units and reasonableness. Is your answer in cm²? Does that speed (150 ms) seem realistic for a bicycle?
Final Revision Tips
Create your own flashcards with the formula on one side and an example on the other.
Practice past paper questions by topic, using the formula sheet as you work.
- Teach someone else how to use a formula. Explaining it solidifies your own understanding.
By learning not just the formulas but their application, you turn a list of symbols into a powerful toolkit for solving any problem the exam throws at you. Good luck!
- Remember: Always check the specification for your specific exam board (AQA, Edexcel, OCR) for any minor variations.