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Year 11 (GCSE) · Formula sheet

Maths formulas

Everything from Year 10 and before, with the Year 11 formulas marked "New".

Area and perimeter

Rectangle: A = l × w
Triangle: A = ½ × b × h
Parallelogram: A = b × h
Trapezium: A = ½(a + b)h
Circle: area: A = πr²
Circle: circumference: C = πd = 2πr

Volume

Cuboid: V = l × w × h
Prism: V = area of cross-section × length
Cylinder: V = πr²h
Units: 1 litre = 1000 cm³1 m³ = 1 000 000 cm³

Pythagoras' theorem and trigonometry

Pythagoras' theorem: a² + b² = c²c is the hypotenuse, the longest side
Sine (SOH): sin θ = oppositehypotenuse
Cosine (CAH): cos θ = adjacenthypotenuse
Tangent (TOA): tan θ = oppositeadjacent

Angles

Angles on a straight line: add up to 180°
Angles around a point: add up to 360°
Angles in a triangle: add up to 180°
Angles in a quadrilateral: add up to 360°
Interior angles of an n-sided polygon: sum = (n − 2) × 180°
Exterior angles of any polygon: add up to 360°
Parallel lines: alternate and corresponding angles are equalco-interior angles add up to 180°

Compound measures

Speed: speed = distancetime
Density: density = massvolume
Pressure: pressure = forcearea

Number

Multiplying powers: aᵐ × aⁿ = aᵐ⁺ⁿ
Dividing powers: aᵐ ÷ aⁿ = aᵐ⁻ⁿ
Power of a power: (aᵐ)ⁿ = aᵐⁿ
Zero and negative powers: a⁰ = 1, a⁻ⁿ = 1aⁿ
Standard form: A × 10ⁿ, where 1 ≤ A < 10
Percentage change: changeoriginal × 100

Graphs and probability

Straight line: y = mx + cm is the gradient, c is the y-intercept
Gradient: m = change in ychange in x
Probability of not happening: P(not A) = 1 − P(A)
Expected frequency: P(A) × number of trials
Independent events: P(A and B) = P(A) × P(B)

Surds

Multiplying: √a × √b = √(ab)
Dividing: √a ÷ √b = √ab
Rationalising: a√b = a√bb
Conjugate pairs: (a + √b)(a − √b) = a² − b

Quadratics

Quadratic formula: x = −b ± √(b² − 4ac)2asolves ax² + bx + c = 0
Completing the square: x² + bx + c = (x + b2)² − (b2)² + c
Difference of two squares: x² − a² = (x + a)(x − a)

Growth and decay

Compound interest: A = P(1 + r100)ⁿ
Depreciation: A = P(1 − r100)ⁿP is the starting amount, r the % rate, n the number of years

Circles, spheres and cones

Arc length: θ360 × 2πr
Sector area: θ360 × πr²
Sphere: V = 43πr³, surface area = 4πr²
Cone: V = 13πr²h, curved surface = πrll is the slant height
Pyramid: V = 13 × base area × h

Similar shapes

Lengths: multiplied by k (the scale factor)
Areas: multiplied by k²
Volumes: multiplied by k³
Congruent triangles: SSS, SAS, ASA, RHS

Circle theorems

Angle at the centre: twice the angle at the circumference
Angle in a semicircle: 90°
Angles in the same segment: are equal
Cyclic quadrilateral: opposite angles add up to 180°
Tangent and radius: meet at 90°

Vectors, statistics and probability

Magnitude of a vector (x, y): √(x² + y²)
Interquartile range: upper quartile − lower quartile
Mutually exclusive events: P(A or B) = P(A) + P(B)

Non-right-angled trianglesNew

Sine rule (sides): asin A = bsin B = csin C
Sine rule (angles): sin Aa = sin Bb = sin Cc
Cosine rule (side): a² = b² + c² − 2bc cos A
Cosine rule (angle): cos A = b² + c² − a²2bc
Area of a triangle: A = ½ab sin CC is the angle between sides a and b

Exact trigonometric valuesNew

sin 0°, 30°, 45°, 60°, 90°: 0, 12, √22, √32, 1
cos 0°, 30°, 45°, 60°, 90°: 1, √32, √22, 12, 0
tan 0°, 30°, 45°, 60°: 0, √33, 1, √3

Algebra and graphsNew

Perpendicular gradients: m₁ × m₂ = −1
Circle centre (0, 0): x² + y² = r²
Iteration: xₙ₊₁ = f(xₙ)
Direct proportion: y = kx or y = kx²
Inverse proportion: y = kx or y = kx²

MotionNew

Velocity-time graph: gradient = acceleration, area = distance
Final velocity: v = u + at
Displacement: s = ut + ½at²
Without time: v² = u² + 2as

Statistics and probabilityNew

Frequency density: frequencyclass width
Either event: P(A or B) = P(A) + P(B) − P(A and B)
Conditional probability: P(A | B) = P(A and B)P(B)