GCSE Maths Formula Sheet – Foundation Tier
Number
- BIDMAS: Brackets → Indices → Division/Multiplication → Addition/Subtraction
- Prime factors: e.g. \(720 = 2^4 \times 3^2 \times 5\)
- Powers & roots: \(a^0 = 1\), \(a^{-n} = \frac{1}{a^n}\)
- Fractions: \(\frac{a}{b} + \frac{c}{d} = \frac{ad+bc}{bd}\), \(\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}\)
- Surds: simplify, e.g. \(\sqrt{80} = 4\sqrt{5}\)
- Standard form: \(a \times 10^n\), where \(1 \leq a < 10\)
Algebra
- Indices laws: \(a^x \cdot a^y = a^{x+y}\), \(\frac{a^x}{a^y} = a^{x-y}\), \((a^x)^y = a^{xy}\)
- Expanding brackets: \((x+a)(x+b) = x^2 + (a+b)x + ab\)
- Difference of squares: \(a^2 – b^2 = (a+b)(a-b)\)
- Quadratics: solve \(x^2 – 8x + 15 = 0 \Rightarrow (x-3)(x-5)=0\)
- Sequences: Arithmetic \(an+d\), Geometric (multiply by constant ratio), Fibonacci (add previous two terms)
Graphs
- Straight line: \(y = mx + c\) (m = gradient, c = intercept)
- Parallel lines: equal gradients
- Standard graphs: \(y = x^2\), \(y = \frac{1}{x}\), \(y = x^3\)
Geometry
- Angles: Straight line = \(180^\circ\), Full turn = \(360^\circ\), Triangle = \(180^\circ\), Polygon = \((n-2)\cdot 180^\circ\), Exterior = \(360^\circ\)
- Pythagoras: \(a^2 + b^2 = c^2\)
- Trigonometry (SOH–CAH–TOA): \(\sin \theta = \frac{\text{opp}}{\text{hyp}}\), \(\cos \theta = \frac{\text{adj}}{\text{hyp}}\), \(\tan \theta = \frac{\text{opp}}{\text{adj}}\)
- Special trig values: \(\sin 30^\circ = \tfrac{1}{2}\), \(\cos 30^\circ = \tfrac{\sqrt{3}}{2}\), \(\tan 30^\circ = \tfrac{1}{\sqrt{3}}\); \(\sin 45^\circ = \tfrac{1}{\sqrt{2}}\), \(\cos 45^\circ = \tfrac{1}{\sqrt{2}}\), \(\tan 45^\circ = 1\); \(\sin 60^\circ = \tfrac{\sqrt{3}}{2}\), \(\cos 60^\circ = \tfrac{1}{2}\), \(\tan 60^\circ = \sqrt{3}\)
Circles
- Circumference = \(\pi d\)
- Area = \(\pi r^2\)
- Arc length = \(\frac{\theta}{360^\circ} \cdot 2\pi r\)
- Sector area = \(\frac{\theta}{360^\circ} \cdot \pi r^2\)
Area & Volume
- Triangle = \(\tfrac{1}{2} \cdot \text{base} \cdot \text{height}\)
- Trapezium = \(\tfrac{1}{2}(a+b)h\)
- Cuboid = \(l \cdot w \cdot h\)
- Cylinder = \(\pi r^2 h\)
- Prism = \(\text{area of cross‑section} \times \text{length}\)
Ratio, Proportion & Percentages
- Ratio: divide total into parts
- Percentages: \(\frac{y}{x} \times 100\%\)
- Speed = \(\frac{\text{distance}}{\text{time}}\)
Probability
- \(P = \frac{\text{favourable outcomes}}{\text{total outcomes}}\)
- Add for mutually exclusive events
- Multiply for independent events
Statistics
- Mean = \(\frac{\text{total}}{\text{number of items}}\)
- Median = middle value
- Mode = most frequent
- Range = largest – smallest
- Correlation: positive, negative, none
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