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Formulae for GCSE

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Full GCSE Mathematics Formula List

This page provides a comprehensive collection of GCSE Maths formulas arranged by topic. It expands beyond the official exam formula sheet and includes key relationships across Number, Algebra, Geometry, Probability and Statistics.

  • Purpose: For revision and reference, not for exam memorisation.
  • Exam Sheet: Visit Formulae Sheet for the official exam version.
  • Interactive: Click Topic Summary for explanations and examples.

 Number

Percentage

Where \( P \) is the principal amount, \( r \) is the interest rate, and \( n \) is the number of times interest is compounded:

\[ A = P\left(1 + \frac{r}{100}\right)^n \]

Index Laws

\[ a^m \cdot a^n = a^{m+n} \]
\[ \frac{a^m}{a^n} = a^{m-n} \]
\[ (a^m)^n = a^{mn} \]
\[ a^0 = 1 \]
\[ a^{-n} = \frac{1}{a^n} \]
\[ a^{\frac{1}{n}} = \sqrt[n]{a} \]

HCF & LCM

Highest Common Factor: Largest number that divides exactly into two or more numbers.

Lowest Common Multiple: Smallest number that is a multiple of two or more numbers.

Standard Form

Write numbers in the form:

\[ a \times 10^n \]

where 1 ≤ a < 10, and n is an integer.

Error

Absolute Error: Difference between measured and true value.

\[ | \text{measured value} - \text{true value} | \]

Percentage Error:

\[ \frac{\text{error}}{\text{true value}} \times 100\% \]

 Algebra

Expanding Brackets

\[ (a + b)^2 = a^2 + 2ab + b^2 \]
\[ (a - b)^2 = a^2 - 2ab + b^2 \]
\[ (a + b)(a - b) = a^2 - b^2 \]

Quadratic formula

The solution of \( ax^2 + bx + c = 0 \)

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

where a ≠ 0

Coordinates and straight line equations

Gradient (slope) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \):

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Equation of a straight line (slope-intercept form):

\[ y = mx + c \]

where \( m \) is the gradient and \( c \) is the y-intercept.

Point-slope form of a line passing through \( (x_1, y_1) \):

\[ y - y_1 = m(x - x_1) \]

Parallel lines: same gradient.

\[ m_1 = m_2 \]

Lines are perpendicular if the product of their gradients is \( -1 \) .

\[ m_1 \cdot m_2 = -1 \]

Midpoint between two points \( (x_1, y_1) \) and \( (x_2, y_2) \):

\[ \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

Distance between two points:

\[ \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Equation of a circle

The standard form of a circle centered at \( (a, b) \) with radius \( r \):

\[ (x - a)^2 + (y - b)^2 = r^2 \]

 Ratio

Compound Measures

\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \]
\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]
\[ \text{Pressure} = \frac{\text{Force}}{\text{Area}} \]

Direct & Inverse Proportion

\[ \text{Direct} = y \propto x \Rightarrow y = kx \]
\[ \text{Inverse} = y \propto \frac{1}{x} \Rightarrow y = \frac{k}{x} \]

 Geometry

Angles properties

Angles on a straight line = \(180^\circ\)
Angles around a point = \(360^\circ\)
Vertically opposite angles are equal
Angles in a triangle = \(180^\circ\)
Angles around a point = \(360^\circ\)

Angles in polygon

Sum of interior angles = \((n - 2) \times 180^\circ\)
Each interior angle of regular polygon = \(\frac{(n - 2) \times 180^\circ}{n}\)
Exterior angle of regular polygon = \(\frac{360^\circ}{n}\)

Circle Theorems

Angle in a semicircle = \( 90^\circ\)
Angles in the same segment are equal
Angle at the centre = \(2 \times \) angle at the circumference
Opposite angles in a cyclic quadrilateral add to \( 180^\circ\)

Pythagoras' Theorem and Trigonometry

In any right-angled triangle:

\( a^2 + b^2 = c^2 \)

In any right-angled triangle ABC where a, b and c are the length of the sides and c is the hypotenuse:

\[ \begin{aligned} \sin A &= \frac{a}{c} \\ \cos A &= \frac{b}{c} \\ \tan A &= \frac{a}{b} \end{aligned} \]

Trigonometry Rules for Non-Right-Angled Triangles

C c A B b a

In any triangle ABC where a, b and c are the length of the sides:

Sine rule

\[ \begin{aligned} \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \end{aligned} \]

Cosine rule

\[ a^2 = b^2 + c^2 - 2bc \cos A \]

Area of triangle

\[ A = \frac{1}{2} ab \sin C \]

Special angles

Angle (°) \( \sin \theta \) \( \cos \theta \) \( \tan \theta \)
30 \( \frac{1}{2} \) \( \frac{\sqrt{3}}{2} \) \( \frac{1}{\sqrt{3}} \)
45 \( \frac{1}{\sqrt{2}} \) \( \frac{1}{\sqrt{2}} \) 1
60 \( \frac{\sqrt{3}}{2} \) \( \frac{1}{2} \) \( \sqrt{3} \)

Perimeter, area and volume

Where a and b are the lengths of the parallel sides and h is their perpendicular separation:

Area of a trapezium

\[ \frac{1}{2}(a + b)h \]

Volume of a prism

\[ \text{Volume} = \text{area of cross section} \times \text{length} \]

Circle area and perimeter

Circumference of a circle

\[ 2\pi r = \pi d \]

Area of a circle

\[ \pi r^2 \]

Sectors of a circle

Where \( r \) is the radius and \( \theta \) is the angle in degrees:

Arc length: \( \frac{\theta}{360} \times 2\pi r \)

Area of sector: \( \frac{\theta}{360} \times \pi r^2 \)

Volumes and surface areas (cylinder, cone, sphere)

Cylinder:

Volume: \( \pi r^2 h \)

Curved surface area: \( 2\pi r h \)

Cone (where \( l \) is the slant height):

Volume: \( \frac{1}{3} \pi r^2 h \)

Curved surface area: \( \pi r l \)

Sphere:

Volume: \( \frac{4}{3} \pi r^3 \)

Surface area (entire): \( 4 \pi r^2 \)

 Probability

Probability

The probability of an event A:

\[ P(A) = \frac{\text{No. of successful outcomes}}{\text{Total no. of outcomes}} \]

Probabilities range from 0 (impossible) to 1 (certain).

\[ 0 \le P(A) \le 1 \]

For mutually exclusive events A and B:

\[ P(A \text{ or } B) = P(A) + P(B) \]

Addition Rule

For any events A and B:

\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

Multiplication Rule

For independent events A and B:

\[ P(A \cap B) = P(A) \times P(B) \]

For dependent events (conditional):

\[ P(A \cap B) = P(A \mid B) \times P(B) \]

 Statistics

Averages

Mean: \( \frac{\sum x}{n} \)

Grouped Mean: \( \frac{\sum fx}{\sum f} \)

Median: Middle value when data is in order

Mode: Most common value

Range

Range = Maximum - Minimum

 Number
  Algebra
  Ratio & Proportion
  Geometry
  Probability
  Statistics
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