GCSE Maths: Formulas to Memorise Flashcards

Question 1

Define Area of a Rectangle.

Accepted Answer

Length multiplied by width, with both measured in the same unit. The answer is in square units, such as cm², because area counts how many unit squares fit inside the shape.

Question 2

Define Area of a Triangle.

Accepted Answer

Half the base multiplied by the perpendicular height, written ½ × b × h. The height must meet the base at a right angle, so a slanted side is never used.

Question 3

Define Area of a Parallelogram.

Accepted Answer

Base multiplied by perpendicular height, with no half. The height is measured at a right angle to the base, not along the slanted side, which is always longer.

Question 4

Define Volume of a Cuboid.

Accepted Answer

Length × width × height. It is the area of the base multiplied by the height, and the answer is in cubic units such as cm³ because volume counts unit cubes.

Question 5

Define Volume of a Cylinder.

Accepted Answer

Volume = πr²h: the area of the circular end, πr², multiplied by the height. A cylinder is a prism with a circle as its cross-section, so the prism rule applies.

Question 6

Define Curved Surface Area of a Cylinder.

Accepted Answer

Curved surface area = 2πrh. The curved face unrolls into a rectangle as wide as the circumference, 2πr, and as tall as the cylinder. Add 2πr² for the two circular ends to get total surface area.

Question 7

Define Arc Length.

Accepted Answer

Arc length = (θ ÷ 360) × 2πr, which is the fraction of the full circumference that the angle θ at the centre cuts off. Use πd in place of 2πr if you are given the diameter.

Question 8

Define Area of a Sector.

Accepted Answer

Sector area = (θ ÷ 360) × πr², which is the fraction of the whole circle's area that is taken by the angle θ at the centre. It uses the same fraction as arc length.

Question 9

Define Interior Angle Sum of a Polygon.

Accepted Answer

The interior angles of any n-sided polygon add up to (n − 2) × 180°. The rule works because one corner can be joined to the others to split the shape into n − 2 triangles.

Question 10

Define Exterior Angle of a Regular Polygon.

Accepted Answer

The exterior angles of any polygon add up to 360°, so each exterior angle of a regular n-sided polygon is 360 ÷ n. The interior angle beside it is 180° minus that.

Question 11

Define Length, Area and Volume Scale Factors.

Accepted Answer

If the lengths of a shape are enlarged by scale factor k, its areas are multiplied by k² and its volumes by k³. Area and volume never scale by k itself.

Question 12

What is Speed?

Accepted Answer

Speed = distance ÷ time, so distance = speed × time and time = distance ÷ speed. Units must match: kilometres and hours give km/h, and metres and seconds give m/s.

Question 13

What is Density?

Accepted Answer

Density = mass ÷ volume. It measures how much mass is packed into each unit of volume, and is usually written in g/cm³ or kg/m³ depending on the units given.

Question 14

What is Pressure?

Accepted Answer

Pressure = force ÷ area. The same force on a smaller area gives a larger pressure, measured in N/m² (pascals) when the force is in newtons and the area is in m².

Question 15

Define Percentage Change.

Accepted Answer

Percentage change = (change ÷ original) × 100, where change is the new value minus the original value. Always divide by the original, never by the new amount.

Question 16

Define Percentage Multiplier.

Accepted Answer

A multiplier turns a percentage change into one multiplication: 1 + rate for an increase and 1 − rate for a decrease. The new amount is the original multiplied by the multiplier.

Question 17

Define Reverse Percentage.

Accepted Answer

A reverse percentage problem gives the amount after a change and asks for the original. Divide the new amount by the multiplier; never take the percentage off the new amount.

Question 18

Define Direct Proportion.

Accepted Answer

Two quantities are in direct proportion when their ratio stays constant: y = kx. Doubling one doubles the other, and the graph is a straight line through the origin.

Question 19

Define Inverse Proportion.

Accepted Answer

Two quantities are in inverse proportion when their product stays constant: y = k ÷ x. Doubling one halves the other, and the graph is a curve that never touches either axis.

Question 20

Define Equation of a Straight Line.

Accepted Answer

y = mx + c is the equation of a straight line, where m is the gradient and c is the y-intercept, the value of y where the line crosses the y-axis.

Question 21

What is Gradient?

Accepted Answer

Gradient = change in y ÷ change in x, found between two points as (y₂ − y₁) ÷ (x₂ − x₁). It measures how steep a line is: rise over run.

Question 22

Define Parallel and Perpendicular Gradients.

Accepted Answer

Parallel lines have equal gradients. Perpendicular lines have gradients that multiply to −1, so the perpendicular gradient is the negative reciprocal of the original (a Higher tier idea).

Question 23

Define Equation of a Circle.

Accepted Answer

x² + y² = r² is the equation of a circle with its centre at the origin (0, 0) and radius r. Every point on the circle satisfies it, and r² is the right-hand side.

Question 24

Define Difference of Two Squares.

Accepted Answer

a² − b² = (a + b)(a − b). Two perfect squares separated by a minus sign factorise into one bracket with a plus and one with a minus, with no middle term.

Question 25

Define Expanding a Squared Bracket.

Accepted Answer

(a + b)² = a² + 2ab + b². Squaring a bracket means multiplying it by itself, so there is a middle term 2ab; writing only a² + b² is the classic error.

Question 26

Define nth Term of a Linear Sequence.

Accepted Answer

For a sequence with common difference d, the nth term is dn + (first term − d). The number in front of n is the difference, and the number added is what is left at term zero.

Question 27

Define Index Laws.

Accepted Answer

Multiply powers of one base by adding indices (aᵐ × aⁿ = aᵐ⁺ⁿ), divide by subtracting them (aᵐ ÷ aⁿ = aᵐ⁻ⁿ), and raise a power to a power by multiplying them ((aᵐ)ⁿ = aᵐⁿ).

Question 28

Define Zero, Negative and Fractional Indices.

Accepted Answer

a⁰ = 1 for any non-zero a, a⁻ⁿ = 1 ÷ aⁿ, and a^(1/n) is the nth root of a. A negative index means a reciprocal, not a negative number.

Question 29

Define Mean from a Frequency Table.

Accepted Answer

Multiply each value by its frequency, add the results, then divide by the total frequency: mean = Σ(f × x) ÷ Σf. Dividing by the number of rows is a common error.

Question 30

Define Position of the Median.

Accepted Answer

For n values in order, the median is in position (n + 1) ÷ 2. When n is even this gives a half-position, so take the mean of the two middle values.

Question 31

Define Interquartile Range.

Accepted Answer

Interquartile range = upper quartile − lower quartile. It is the spread of the middle half of the data and, unlike the range, is not distorted by one extreme value.

Question 32

Define Relative Frequency.

Accepted Answer

Relative frequency = frequency of the outcome ÷ total number of trials. It is an experimental estimate of probability, and it becomes more reliable as the number of trials grows.

Question 33

Define Expected Frequency.

Accepted Answer

Expected number of outcomes = probability × number of trials. It is an average prediction over many trials, not a guarantee of what will happen in any one set of trials.

Question 34

Define Probability of an Event Not Happening.

Accepted Answer

P(not A) = 1 − P(A). The probabilities of all possible outcomes add up to 1, so the chance an event does not happen is what is left after taking away the chance it does.

Question 35

Define Independent Events Rule.

Accepted Answer

For independent events, P(A and B) = P(A) × P(B), because one event happening does not change the chance of the other. On a tree diagram you multiply along the branches.

Question 36

Define Frequency Density.

Accepted Answer

Frequency density = frequency ÷ class width. A histogram plots it as the bar height, so the area of each bar, not its height, represents the frequency (a Higher tier topic).

Question 37

Define Exact Trig Values at 30° and 60°.

Accepted Answer

sin 30° = cos 60° = ½, sin 60° = cos 30° = √3 ÷ 2, tan 30° = 1 ÷ √3 and tan 60° = √3. These exact values are not printed on the formulae sheets.

Question 38

Define Exact Trig Values at 0°, 45° and 90°.

Accepted Answer

sin 0° = 0, cos 0° = 1 and tan 0° = 0. At 45° sine and cosine both equal √2 ÷ 2 and tan is 1. sin 90° = 1, cos 90° = 0, and tan 90° is undefined.