GCSE Maths Formula Sheet
All the formulas you need for your GCSE Maths exams - Foundation & Higher
AQA Exam Information
Some formulas are given in the exam (marked with a green badge below). However, knowing them by heart saves time and reduces errors. All other formulas must be memorised.
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Algebra
Quadratic Formula
Given
H
x = −b ± √(b² − 4ac)2a
Use when: Solving ax² + bx + c = 0 when you can't factorise easily.
Remember: Check the discriminant b² − 4ac first to see how many solutions.
Remember: Check the discriminant b² − 4ac first to see how many solutions.
Difference of Two Squares
F
H
a² − b² = (a + b)(a − b)
Use when: Factorising expressions like x² − 9 or 4x² − 25.
Tip: Both terms must be perfect squares with a minus between them.
Tip: Both terms must be perfect squares with a minus between them.
Equation of a Straight Line
F
H
y = mx + c
m = gradient (slope), c = y-intercept (where line crosses y-axis).
Tip: Parallel lines have the same gradient.
Tip: Parallel lines have the same gradient.
nth Term (Linear Sequence)
F
H
nth term = dn + (a − d)
d = common difference, a = first term.
Example: 3, 7, 11, 15... → d=4, a=3, so 4n + (3−4) = 4n − 1
Example: 3, 7, 11, 15... → d=4, a=3, so 4n + (3−4) = 4n − 1
Geometry - Area & Perimeter
Area of Rectangle
F
H
Area = length × width
Units: If length is in cm, area is in cm².
Area of Triangle
F
H
Area = 12 × base × height
Remember: Height must be perpendicular (90°) to the base.
Area of Trapezium
Given
F
H
Area = 12(a + b) × h
a and b are the parallel sides, h is the perpendicular height.
Area of Parallelogram
F
H
Area = base × height
Not base × slant side! Height is perpendicular to the base.
Circles
Circumference of Circle
F
H
C = πd = 2πr
d = diameter, r = radius. Remember: d = 2r
Area of Circle
F
H
A = πr²
Warning: Use radius, not diameter! If given diameter, divide by 2 first.
Arc Length
H
Arc = θ360 × πd
θ = angle at centre. It's just a fraction of the full circumference.
Area of Sector
H
Sector = θ360 × πr²
θ = angle at centre. It's just a fraction of the full circle area.
3D Shapes - Volume & Surface Area
Volume of Cuboid
F
H
V = length × width × height
Units: If lengths are in cm, volume is in cm³.
Volume of Prism
F
H
V = cross-section area × length
Works for any prism: triangular, L-shaped, etc. Find the end area first!
Volume of Cylinder
Given
F
H
V = πr²h
A cylinder is just a circular prism: circle area × height.
Volume of Cone
Given
H
V = 13πr²h
A cone is ⅓ of a cylinder with the same base and height.
Volume of Pyramid
Given
H
V = 13 × base area × height
Works for any pyramid. A cone is just a pyramid with a circular base!
Volume of Sphere
Given
H
V = 43πr³
Tip: r³ not r²! Cube the radius.
Surface Area of Sphere
Given
H
SA = 4πr²
It's exactly 4 times the area of a circle with the same radius.
Curved Surface Area of Cone
Given
H
CSA = πrl
l = slant height (not vertical height!). Use Pythagoras if needed: l² = r² + h²
Pythagoras & Trigonometry
Pythagoras' Theorem
F
H
a² + b² = c²
c = hypotenuse (longest side, opposite the right angle).
Finding shorter side: a² = c² − b²
Finding shorter side: a² = c² − b²
SOH CAH TOA
F
H
sin θ = OH
cos θ = AH
tan θ = OA
O = Opposite, A = Adjacent, H = Hypotenuse.
Label sides first, then choose the right ratio!
Label sides first, then choose the right ratio!
Sine Rule
Given
H
asin A =
bsin B =
csin C
Use when: You have a side-angle pair and need another side or angle.
Flip it to sin Aa when finding an angle.
Flip it to sin Aa when finding an angle.
Cosine Rule
Given
H
a² = b² + c² − 2bc cos A
Use when: You have all 3 sides, OR 2 sides and the included angle.
To find angle: cos A = b² + c² − a²2bc
To find angle: cos A = b² + c² − a²2bc
Area of Triangle (Trig)
Given
H
Area = 12ab sin C
Use when: You have 2 sides and the angle between them.
No need to calculate the height!
No need to calculate the height!
Exact Trig Values
F
H
| θ | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin | 0 | ½ | √22 | √32 | 1 |
| cos | 1 | √32 | √22 | ½ | 0 |
| tan | 0 | 1√3 | 1 | √3 | − |
Must memorise! Often tested in non-calculator papers.
Compound Measures
Speed, Distance, Time
F
H
Speed = DistanceTime
Triangle: D on top, S × T on bottom.
D = S × T | T = D ÷ S
D = S × T | T = D ÷ S
Density
F
H
Density = MassVolume
Units: g/cm³ or kg/m³
M = D × V | V = M ÷ D
M = D × V | V = M ÷ D
Pressure
F
H
Pressure = ForceArea
Units: N/m² (Pascals) or N/cm²
F = P × A | A = F ÷ P
F = P × A | A = F ÷ P
Compound Interest / Growth
F
H
Final = Original × (multiplier)ⁿ
Multiplier: +5% → 1.05, −3% → 0.97
n = number of years/periods
n = number of years/periods
Probability & Statistics
Probability
F
H
P(event) = favourable outcomestotal outcomes
Range: Always between 0 (impossible) and 1 (certain).
P(not A) = 1 − P(A)
P(not A) = 1 − P(A)
Mean
F
H
Mean = sum of valuesnumber of values
From table: Σfx ÷ Σf (sum of frequency × value, divided by total frequency)
Frequency Density (Histograms)
H
FD = FrequencyClass Width
Reading histograms: Frequency = FD × Class Width
Area of bar = Frequency
Area of bar = Frequency
Angles & Polygons
Sum of Interior Angles
F
H
Sum = 180° × (n − 2)
n = number of sides.
Triangle: 180°, Quad: 360°, Pentagon: 540°, Hexagon: 720°
Triangle: 180°, Quad: 360°, Pentagon: 540°, Hexagon: 720°
Interior Angle (Regular Polygon)
F
H
Interior = 180(n − 2)n
Or: Interior = 180° − Exterior
Exterior Angles
F
H
Exterior angles always sum to 360°
Regular polygon: Each exterior = 360° ÷ n
Interior + Exterior = 180°
Interior + Exterior = 180°