Selective School Formulas

Area of a Rectangle

Area = length × width

Area of a Triangle

Area = 1/2 × base × height

Area of a Circle

Area = π × r²

Area of a Trapezium

Area = 1/2 × (a + b) × height, where a and b are parallel sides

Perimeter of a Rectangle

Perimeter = 2 × (length + width)

Circumference of a Circle

Circumference = 2πr or π × diameter

Volume of a Rectangular Prism

Volume = length × width × height

Volume of a Cylinder

Volume = π × r² × height

Volume of a Cone

Volume = (1/3) × π × r² × height

Volume of a Sphere

Volume = (4/3) × π × r³

Surface Area of a Rectangular Prism

SA = 2lw + 2lh + 2wh

Surface Area of a Cylinder

SA = 2πr² + 2πrh

Surface Area of a Sphere

SA = 4πr²

Surface Area of a Cone

SA = πr² + πrl

Pythagoras' Theorem

a² + b² = c² (in a right-angled triangle)

Speed Formula

Speed = Distance ÷ Time

Distance Formula

Distance = √[(x₂ − x₁)² + (y₂ − y₁)²]

Time Formula

Time = Distance ÷ Speed

Midpoint Formula

Midpoint = ((x₁ + x₂)/2 , (y₁ + y₂)/2)

Linear Equation from Two Points

y − y₁ = m(x − x₁), where m = (y₂ − y₁)/(x₂ − x₁)

Linear Equation Form

y = mx + c

Gradient Formula

m = (y₂ − y₁)/(x₂ − x₁)

Solving Simultaneous Equations

Use substitution or elimination method

Quadratic Equation Standard Form

y = ax² + bx + c

Axis of Symmetry (Parabola)

x = −b ÷ 2a

Vertex of a Parabola

Vertex = (−b/2a , f(−b/2a))

Quadratic Formula

x = [−b ± √(b² − 4ac)] ÷ 2a

Perfect Squares

a² + 2ab + b² = (a + b)²

Difference of Two Squares

a² − b² = (a + b)(a − b)

Binomial Product

(a + b)(c + d) = ac + ad + bc + bd

Solving Inequalities

Treat like equations, but reverse the sign when multiplying/dividing by a negative

Algebraic Fractions

Add or subtract: Use common denominators. Multiply: Multiply tops and bottoms

Factorising Quadratic Trinomials

ax² + bx + c = (px + q)(rx + s), where p × r = a, q × s = c

Expanding Brackets

a(b + c) = ab + ac

Factoring Common Terms

ab + ac = a(b + c)

Probability Formula

P = Favourable Outcomes ÷ Total Outcomes

Complementary Probability

P(not A) = 1 − P(A)

Expected Value

E = Probability × Value

Compound Interest Formula

A = P(1 + r/n)^(nt)

Simple Interest Formula

I = PRT

Ratios (Simplify)

Divide both parts by common factors

Rates

Rate = Quantity ÷ Time (e.g., km/h, $/kg)

Sine Ratio

sin(θ) = opposite ÷ hypotenuse

Cosine Ratio

cos(θ) = adjacent ÷ hypotenuse

Tangent Ratio

tan(θ) = opposite ÷ adjacent

SOHCAHTOA

Memory aid: sin = o/h, cos = a/h, tan = o/a

Trigonometry: Finding a side

Rearrange SOHCAHTOA depending on known angle

Trigonometry: Finding an angle

Use inverse functions: θ = sin⁻¹(o/h), cos⁻¹(a/h), tan⁻¹(o/a)

Sum of Angles in a Triangle

180°

Exterior Angle of a Triangle

Exterior Angle = sum of two opposite interior angles

Angles on a Straight Line

180°

Angles Around a Point

360°

Angles in Parallel Lines

Alternate angles = equal, Corresponding angles = equal, Co-interior = supplementary

Sum of Interior Angles in a Polygon

(n − 2) × 180°

Sum of Exterior Angles in a Polygon

360°

Mean

Mean = Sum of values ÷ Number of values

Median

Middle number when values are ordered

Mode

Most frequent value

Range

Largest value − smallest value