Every formula, one page — USS revision

What this page is for

Nothing here is explained; the topic pages do that. This is the page to open the night before, or in the ten minutes before the paper, when what you need is the formula and not the reasoning behind it. Units are included on purpose: they are the fastest way to catch the wrong formula.

Mensuration — flat shapes

Shape	Formula	Unit
Square perimeter	4 × side	cm
Square area	side²	cm²
Rectangle perimeter	2(l + b)	cm
Rectangle area	l × b	cm²
Triangle area	½ × base × height	cm²
Parallelogram area	base × height	cm²
Trapezium area	½ × (a + b) × height	cm²
Circle circumference	2πr	cm
Circle area	πr²	cm²

Mensuration — solids

Solid	Formula	Unit
Cube volume	side³	cm³
Cube surface area	6 × side²	cm²
Cuboid volume	l × b × h	cm³
Cuboid surface area	2(lb + bh + hl)	cm²
Cylinder volume	πr²h	cm³

Angles and polygons

Triangle angle sum = 180°. Quadrilateral = 360°.
Interior angle sum of an n-sided polygon = (n − 2) × 180°.
Exterior angles of any polygon add to 360°.
Each interior angle of a regular n-gon = (n − 2) × 180 ÷ n.
Complementary add to 90°; supplementary add to 180°.
Pythagoras: a² + b² = c² (c is the hypotenuse).

Arithmetic

x% of N = N × x/100
Profit% = (profit ÷ cost price) × 100 — always on the cost price.
Loss% = (loss ÷ cost price) × 100
Percentage increase = (increase ÷ original) × 100
Simple interest = P × R × T ÷ 100. Amount = P + SI.
Mean = total ÷ count, and total = mean × count.
Range = largest − smallest
HCF × LCM = product of the two numbers
Ratio a : b → total is a + b parts. Find one part first.

Algebra identities and index laws

Rule	Note
(a + b)² = a² + 2ab + b²	the 2ab is the term people drop
(a − b)² = a² − 2ab + b²	middle term negative
a² − b² = (a + b)(a − b)	difference of two squares
aᵐ × aⁿ = aᵐ⁺ⁿ	indices add
aᵐ ÷ aⁿ = aᵐ⁻ⁿ	indices subtract
(aᵐ)ⁿ = aᵐⁿ	indices multiply
(ab)ⁿ = aⁿbⁿ	the power hits both parts
a⁰ = 1	any non-zero a

Numbers worth knowing on sight

Family	Values
Squares	1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144
Cubes	1, 8, 27, 64, 125, 216
Primes under 30	2, 3, 5, 7, 11, 13, 17, 19, 23, 29
Fibonacci	1, 1, 2, 3, 5, 8, 13, 21, 34
Triangular	1, 3, 6, 10, 15, 21, 28
Fraction → %	½ = 50 · ¼ = 25 · ¾ = 75 · ⅕ = 20 · ⅛ = 12.5 · ⅓ = 33⅓

Science — units and quantities

Quantity	Unit	Symbol
Force	newton	N
Work / energy	joule	J
Power	watt	W
Pressure	pascal	Pa
Current	ampere	A
Potential difference	volt	V
Resistance	ohm	Ω
Temperature	degree Celsius	°C

Science — relationships and numbers

Speed = distance ÷ time
Density = mass ÷ volume
Force = mass × acceleration
Speed of light ≈ 3 × 10⁸ m/s. Speed of sound in air ≈ 340 m/s.
Air: 78% nitrogen, 21% oxygen.
Water boils at 100 °C, freezes at 0 °C (at sea level).
Clinical thermometer range: 35 °C to 42 °C.
Human heart: 4 chambers. Sense organs: 5.
pH: below 7 acidic, 7 neutral, above 7 basic.

Science — reactions to remember

Reaction	Product
Acid + base	salt + water
Acid + metal	salt + hydrogen (burns with a pop)
Acid + carbonate	salt + water + carbon dioxide
CO₂ + lime water	milky (the standard CO₂ test)
Acid on blue litmus	turns it red
Base on red litmus	turns it blue
Photosynthesis	CO₂ + water → glucose + oxygen
Respiration	oxygen + food → energy + CO₂ + water

Symbols and formulas

Name	Symbol
Gold	Au
Silver	Ag
Iron	Fe
Sodium	Na
Potassium	K
Copper	Cu
Water	H₂O
Common salt	NaCl
Baking soda	NaHCO₃
Carbon dioxide	CO₂
Limestone	CaCO₃
Brass	copper + zinc
Bronze	copper + tin
Steel	iron + carbon

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