GCSE & IGCSE Mathematics key terms

176 terms with short definitions you can learn for the exam. Type to filter the list.

Key terms 101

Acute angle: An angle less than 90°.
Algebraic fraction: A fraction with an algebraic expression on the top, the bottom or both, such as x + 1/x − 3.
Alternate angles: Angles on opposite sides of a line that crosses two parallel lines, inside the parallel lines. They are equal (a Z shape).
Alternate segment theorem: The angle between a tangent and a chord equals the angle in the alternate segment.
Arc: Part of the circumference of a circle.
Arithmetic sequence: A sequence where you add the same number (the common difference) each time, such as 5, 8, 11, 14.
Bearing: A direction measured clockwise from north, written with three figures, such as 045°.
Bias: When a sample, survey or experiment does not fairly represent the whole population, or when outcomes are not equally likely.
Bounds: The smallest (lower bound) and largest (upper bound) values a rounded number could have been.
Box plot: A diagram showing the minimum, lower quartile, median, upper quartile and maximum of a data set.
Chord: A straight line joining two points on a circle.
Coefficient: The number in front of a letter in an algebraic term. In 7x^{2}, the coefficient is 7.
Completing the square: Writing x^{2} + bx + c in the form (x + p)^{2} + q, which shows the turning point.
Composite function: A function made by applying one function then another. fg(x) means apply g first, then f.
Compound interest: Interest paid on the original amount and on interest already added, so the amount grows by the same percentage each period.
Conditional probability: The probability of an event given that another event has happened, such as P(B given A).
Congruent: Exactly the same shape and size.
Corresponding angles: Angles in matching positions where a line crosses two parallel lines. They are equal (an F shape).
Cosine rule: a^{2} = b^{2} + c^{2} − 2bc cos A, used in any triangle when you know two sides and the angle between them, or all three sides.
Cumulative frequency: A running total of frequencies, used to estimate the median and quartiles.
Cyclic quadrilateral: A quadrilateral with all four vertices on a circle. Opposite angles add to 180°.
Density: Mass per unit volume: density = mass ÷ volume, for example in g/cm^{3}.
Depreciation: A fall in value over time, often by a fixed percentage each year.
Derivative IGCSE: The gradient function dy/dx of a curve. For y = x^{n}, dy/dx = nx^{n − 1}.
Direct proportion: Two quantities are in direct proportion when one is a constant multiple of the other: y = kx.
Enlargement: A transformation that changes size by a scale factor from a centre of enlargement. Angles stay the same.
Equation: A statement with an equals sign that is true for particular values of the letter, such as 3x + 1 = 10.
Error interval: The range of values a rounded or truncated number could have taken, written as an inequality such as 6.5 ≤ x < 7.5.
Estimate: An approximate answer, usually found by rounding each number to 1 significant figure.
Expression: A collection of terms with no equals sign, such as 4x − 3y.
Exterior angle: The angle between one side of a polygon and the extension of the next side. The exterior angles of any polygon add to 360°.
Factorise: Write an expression as a product by taking out common factors or putting it into brackets.
Formula: A rule linking two or more variables, such as A = πr^{2}.
Frequency density: Frequency ÷ class width. It is the height of each bar in a histogram.
Function: A rule that turns each input into one output, written f(x).
Gradient: The steepness of a line: change in y ÷ change in x. It is m in y = mx + c.
Highest common factor (HCF): The largest number that divides exactly into two or more numbers.
Histogram: A chart for grouped continuous data where the area of each bar represents frequency.
Hypotenuse: The longest side of a right-angled triangle, opposite the right angle.
Identity: A statement true for every value of the variable, written with ≡, such as 2(x + 3) ≡ 2x + 6.
Independent events: Events where one happening does not change the probability of the other. P(A and B) = P(A) × P(B).
Index (power): The small number showing how many times a base is multiplied by itself. In 5^{3}, the index is 3.
Inequality: A statement using <, >, ≤ or ≥ to compare two quantities.
Integer: A whole number: positive, negative or zero.
Interquartile range (IQR): Upper quartile minus lower quartile. It measures the spread of the middle half of the data.
Intersection (∩): The elements that are in both sets.
Inverse function: The function that undoes f, written f^{−1}(x).
Inverse proportion: As one quantity increases, the other decreases so that their product is constant: y = k/x.
Irrational number: A number that cannot be written as a fraction of two integers, such as √2 or π.
Iteration GCSE: Repeating a process, feeding each answer back in, to get closer to the solution of an equation: x_{n + 1} = f(x_{n}).
Lowest common multiple (LCM): The smallest number that is a multiple of two or more numbers.
Mean: The total of the values divided by the number of values.
Median: The middle value when the data are in order.
Mode: The value (or class) that occurs most often.
Mutually exclusive: Events that cannot happen at the same time. P(A or B) = P(A) + P(B).
nth term: A rule that gives any term of a sequence from its position n, such as 3n + 2.
Obtuse angle: An angle between 90° and 180°.
Parallel lines: Lines that never meet. They have the same gradient.
Perpendicular lines: Lines that meet at 90°. Their gradients multiply to −1.
Pressure: Force per unit area: pressure = force ÷ area, for example in N/m^{2}.
Prime factor: A factor that is a prime number. Every whole number above 1 can be written as a product of prime factors.
Prime number: A whole number with exactly two factors: 1 and itself. 1 is not prime.
Prism: A 3D shape with the same cross-section all the way through.
Product: The result of multiplying.
Product rule for counting GCSE: If one choice can be made in m ways and a second in n ways, together they can be made in m × n ways.
Proof: A logical argument showing a statement is always true, using algebra that works for every case.
Quadratic: An expression or equation where the highest power of x is 2, such as x^{2} − 5x + 6.
Quartiles: The values a quarter (lower quartile) and three quarters (upper quartile) of the way through ordered data.
Radius: The distance from the centre of a circle to its edge.
Range: The largest value minus the smallest value. It measures spread.
Ratio: A comparison of two or more quantities, written a : b.
Rationalise the denominator: Rewrite a fraction so the denominator has no surd, for example 1/√3 = √3/3.
Reciprocal: 1 divided by a number. The reciprocal of 4 is 1/4; a number times its reciprocal is 1.
Recurring decimal: A decimal with a digit or block of digits that repeats forever, such as 0.181818...
Reflection: A transformation that flips a shape in a mirror line.
Relative frequency: Number of times an outcome happens ÷ number of trials. It estimates probability.
Roots of an equation: The solutions. For a quadratic graph, the x-coordinates where it crosses the x-axis.
Rotation: A transformation that turns a shape through an angle, in a direction, about a centre of rotation.
Sample space: A list or table of all the possible outcomes.
Scale factor: The number every length is multiplied by in an enlargement or between similar shapes.
Sector: The region of a circle between two radii and an arc, like a slice of pizza.
Segment: The region of a circle between a chord and an arc.
Significant figures: The digits of a number counted from the first non-zero digit.
Similar: The same shape but a different size: angles are equal and lengths are in the same ratio.
Simultaneous equations: Two equations with two unknowns that are true at the same time.
Sine rule: a/sin A = b/sin B = c/sin C, used in any triangle when you know a side and its opposite angle.
Standard form: A number written as A × 10^{n} where 1 ≤ A < 10 and n is an integer.
Stationary point IGCSE: A point on a curve where the gradient is zero: a maximum or a minimum turning point.
Subject of a formula: The single variable on its own on one side, such as v in v = u + at.
Surd: A root that cannot be simplified to a whole number, such as √5, left in exact form.
Tangent (to a circle): A straight line that touches a circle at exactly one point. It meets the radius at 90°.
Term: A single number, letter or product of them in an expression, such as 5x or −3.
Translation: A transformation that slides a shape without turning it, described by a column vector.
Tree diagram: A diagram of branches showing the outcomes of two or more events and their probabilities.
Truncate: Cut a number off at a certain point without rounding. 3.78 truncated to 1 decimal place is 3.7.
Turning point: The point where a curve changes from going down to going up or the other way: a minimum or maximum.
Union (∪): The elements in either set or both.
Universal set (ξ): The set of everything being considered in a problem.
Vector: A quantity with size and direction, written as a column vector or in bold, such as a.
Venn diagram: Overlapping circles inside a rectangle that show sets and how they overlap.
Vertically opposite angles: The equal angles opposite each other where two straight lines cross.

Formulae on the exam aid sheet (2026 and 2027 exams) 13

Area of a circle GCSE: πr^{2}
Area of a trapezium GCSE: 1/2(a + b)h, where a and b are the parallel sides and h is the perpendicular height
Area of a triangle (Higher sheet only) GCSE: 1/2ab sin C
Circumference of a circle GCSE: 2πr = πd
Compound interest GCSE: Total accrued = P(1 + r/100)^{n}, where P is the principal, r the interest rate per period and n the number of periods
Cosine rule (Higher sheet only) GCSE: a^{2} = b^{2} + c^{2} − 2bc cos A
Probability: A and B (Higher sheet only) GCSE: P(A and B) = P(A given B) × P(B)
Probability: A or B GCSE: P(A or B) = P(A) + P(B) − P(A and B)
Pythagoras' theorem GCSE: a^{2} + b^{2} = c^{2}, where c is the hypotenuse
Quadratic formula (Higher sheet only) GCSE: For ax^{2} + bx + c = 0, a ≠ 0: x = −b ± √{b^{2} − 4ac}/2a
Sine rule (Higher sheet only) GCSE: a/sin A = b/sin B = c/sin C
Trigonometric ratios GCSE: sin A = a/c, cos A = b/c, tan A = a/b (a opposite, b adjacent, c hypotenuse)
Volume of a prism GCSE: area of cross-section × length

Formulae always given in the question 5

Curved surface area of a cone GCSE: πrl, where l is the slant height
Kinematics formulae GCSE: v = u + at, s = ut + 1/2at^{2}, v^{2} = u^{2} + 2as (u initial velocity, v final velocity, a constant acceleration, s displacement, t time)
Surface area of a sphere GCSE: 4πr^{2}
Volume of a cone GCSE: 1/3πr^{2}h
Volume of a sphere GCSE: 4/3πr^{3}

Formulae you must learn (not on any sheet) 23

Arc length GCSE: θ/360 × 2πr
Area of a parallelogram GCSE: base × perpendicular height
Area of a rectangle GCSE: length × width
Area of a sector GCSE: θ/360 × πr^{2}
Area of a triangle GCSE: 1/2 × base × perpendicular height
Density GCSE: density = mass ÷ volume
Equation of a circle centre the origin (Higher) GCSE: x^{2} + y^{2} = r^{2}
Equation of a straight line GCSE: y = mx + c (m gradient, c y-intercept)
Exact trigonometric values GCSE: sin 30° = cos 60° = 1/2; sin 60° = cos 30° = √3/2; sin 45° = cos 45° = √2/2; tan 45° = 1; tan 30° = √3/3; tan 60° = √3; sin 0° = cos 90° = tan 0° = 0; sin 90° = cos 0° = 1
Expected frequency GCSE: probability × number of trials
Exterior angle of a regular polygon GCSE: 360° ÷ n
Frequency density (Higher) GCSE: frequency ÷ class width
Gradient of a line GCSE: m = y_{2} − y_{1}/x_{2} − x_{1}
Laws of indices GCSE: a^{m} × a^{n} = a^{m + n}; a^{m} ÷ a^{n} = a^{m − n}; (a^{m})^{n} = a^{mn}; a^{0} = 1; a^{−n} = 1/a^{n}; a^{1/n} is the nth root of a
Percentage change GCSE: change/original × 100
Perpendicular gradients GCSE: m_{1} × m_{2} = −1
Pressure GCSE: pressure = force ÷ area
Probabilities of all outcomes GCSE: They add up to 1, so P(not A) = 1 − P(A)
Simple interest GCSE: interest = principal × rate × time ÷ 100
Speed GCSE: speed = distance ÷ time
Sum of interior angles of a polygon GCSE: (n − 2) × 180°
Volume of a cuboid GCSE: length × width × height
Volume of a cylinder GCSE: πr^{2}h (a prism with a circular cross-section)

Formulae on the formulae sheet 13

Area of a trapezium IGCSE: 1/2(a + b)h
Area of a triangle (Higher sheet only) IGCSE: 1/2ab sin C
Cosine rule (Higher sheet only) IGCSE: a^{2} = b^{2} + c^{2} − 2bc cos A
Curved surface area of a cone (Higher sheet only) IGCSE: πrl
Curved surface area of a cylinder IGCSE: 2πrh
Quadratic formula (Higher sheet only) IGCSE: For ax^{2} + bx + c = 0, a ≠ 0: x = −b ± √{b^{2} − 4ac}/2a
Sine rule (Higher sheet only) IGCSE: a/sin A = b/sin B = c/sin C
Sum of an arithmetic series (Higher sheet only) IGCSE: S_{n} = n/2[2a + (n − 1)d]
Surface area of a sphere (Higher sheet only) IGCSE: 4πr^{2}
Volume of a cone (Higher sheet only) IGCSE: 1/3πr^{2}h
Volume of a cylinder IGCSE: πr^{2}h
Volume of a prism IGCSE: area of cross-section × length
Volume of a sphere (Higher sheet only) IGCSE: 4/3πr^{3}

Formulae you must learn (not on the sheet) 21

Arc length and sector area IGCSE: θ/360 × 2πr and θ/360 × πr^{2}
Area of a circle IGCSE: πr^{2}
Area of a parallelogram IGCSE: base × perpendicular height
Area of a rectangle IGCSE: length × width
Area of a triangle IGCSE: 1/2 × base × perpendicular height
Circumference of a circle IGCSE: 2πr = πd
Compound interest and depreciation IGCSE: amount = original × (1 ± r/100)^{n}
Differentiation (Higher) IGCSE: If y = ax^{n}, then dy/dx = nax^{n − 1}; stationary points where dy/dx = 0
Exterior angle of a regular polygon IGCSE: 360° ÷ n
Frequency density (Higher) IGCSE: frequency ÷ class width
Gradient and straight line IGCSE: m = y_{2} − y_{1}/x_{2} − x_{1}; y = mx + c
Laws of indices IGCSE: a^{m} × a^{n} = a^{m + n}; a^{m} ÷ a^{n} = a^{m − n}; (a^{m})^{n} = a^{mn}; a^{0} = 1; a^{−n} = 1/a^{n}
nth term of an arithmetic sequence IGCSE: a + (n − 1)d
Percentage change IGCSE: change/original × 100
Perpendicular gradients (Higher) IGCSE: m_{1} × m_{2} = −1
Probability rules IGCSE: P(not A) = 1 − P(A); mutually exclusive: P(A or B) = P(A) + P(B); independent: P(A and B) = P(A) × P(B)
Pythagoras' theorem IGCSE: a^{2} + b^{2} = c^{2}, where c is the hypotenuse
Speed, density, pressure IGCSE: speed = distance ÷ time; density = mass ÷ volume; pressure = force ÷ area
Sum of interior angles of a polygon IGCSE: (n − 2) × 180°
Trigonometric ratios IGCSE: sin x = opp/hyp, cos x = adj/hyp, tan x = opp/adj
Velocity and acceleration from displacement (Higher) IGCSE: v = ds/dt, a = dv/dt

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