S1 · 10. Basic Geometry

Primes, HCF and LCM S1 · 1

Two numbers, their HCF and LCM

a \times b = HCF \times LCM

two numbers only — it fails for three

Integers, rational and real numbers S1 · 2

Signs when multiplying or dividing	same signs → positive · different signs → negative
Order of operations	brackets → powers and roots → × and ÷ (left to right) → + and − (left to right)−4² = −16, but (−4)² = 16

Basic algebra S1 · 4

The Distributive Law	$a (b + c) = a b + a c$
A minus sign in front of a bracket	$- (x + y) = - x - y$ every sign inside flips

Linear equations S1 · 5

Cross-multiplying

\frac{a}{b} = \frac{c}{d}

→

a d = b c

Linear functions and graphs S1 · 6

Equation of a straight line	$y = m x + c$ m is the gradient, c the y-intercept (the syllabus writes y = ax + b)
Gradient	$\frac{vertical change}{horizontal change} = \frac{rise}{run}$ go from the left-hand point to the right-hand one
Horizontal and vertical lines	y = k has gradient 0 · x = k has an undefined gradient

Number patterns S1 · 7

General term of a linear sequence	$T_{n} = d n + c$ d is the common difference and c = T₁ − d
The nth triangular number	$T_{n} = \frac{n (n + 1)}{2}$

Percentage S1 · 8

A percentage	$x % = \frac{x}{100}$ and 100% = 1
Percentage change	$\frac{increase or decrease}{original value} \times 100 %$ always divide by the original value
New value after a change	new value = final percentage × original valuee.g. a 15% rise: 115% × original
Profit or loss as a percentage	$\frac{profit or loss}{cost price} \times 100 %$ unless the question names the selling price
Percentage discount	$\frac{discount}{marked price} \times 100 %$

Ratio and rate S1 · 9

Simple interest	$I = \frac{P R T}{100}$ P principal, R% rate per year, T years
Speed	$speed = \frac{distance}{time}$ so distance = speed × time and time = distance ÷ speed
Average speed	$\frac{total distance travelled}{total time taken}$ not the average of the speeds
Changing units of speed	km/h ÷ 3.6 = m/s · m/s × 3.6 = km/h

Basic geometry S1 · 10

Angles on a straight line · at a point	add up to 180° (adj. ∠s on a str. line) · add up to 360° (∠s at a point)
Vertically opposite angles	are equal (vert. opp. ∠s)
Parallel lines cut by a transversal	corresponding angles are equal · alternate angles are equal · interior angles add up to 180°write the reason in brackets on every line

Polygons and geometrical constructions S1 · 11

Angles of a triangle	add up to 180° · an exterior angle = the sum of the two interior opposite angles
Sum of the interior angles of an n-sided polygon	$(n - 2) \times 180 °$
Sum of the exterior angles of any polygon	360°and at each vertex, interior angle + exterior angle = 180°
Regular polygon: each angle	$exterior = \frac{360 °}{n}$ $interior = \frac{(n - 2) \times 180 °}{n}$

Perimeter and area of plane figures S1 · 12

Circle	$circumference = 2 π r = π d$ $area = π r^{2}$ halve a diameter before it goes into either formula
Parallelogram	$area = b h$ h is perpendicular to the base b, not the slanting side
Trapezium	$area = \frac{1}{2} (a + b) h$ a and b are the parallel sides
Triangle	$area = \frac{1}{2} b h$ with the perpendicular height

Volume and surface area of prisms and cylinders S1 · 13

Prism (cuboid, cube, any cross-section)	volume = base area × height · total surface area = perimeter of base × height + 2 × base areacuboid: V = lbh, surface 2(lb + lh + bh); cube: l³ and 6l²
Cylinder	$V = π r^{2} h$ $curved surface = 2 π r h$ closed cylinder total: 2πr² + 2πrh
Units	1 m² = 10 000 cm² · 1 m³ = 1 000 000 cm³ · 1 ml = 1 cm³ · 1 litre = 1000 cm³

Statistical data handling S1 · 14

Pie chart sector angle

\frac{frequency of the category}{total frequency} \times 360 °

Linear graphs and simultaneous equations S2 · 1

A linear equation in two variables	ax + by = k is a straight linemake y the subject to read off its gradient
How many solutions?	different gradients → one · same gradient, different intercept → none · the same line → infinitely many

Linear inequalities S2 · 2

Multiplying or dividing by a negative number

reverses the inequality signevery other move is the same as for an equation

Expansion and factorisation S2 · 3

Two brackets	$(a + b) (c + d) = a c + a d + b c + b d$
Factorising x² + bx + c	$(x + m) (x + n)$ where m × n = c and m + n = b

Special algebraic identities S2 · 4

Perfect squares	${(a + b)}^{2} = a^{2} + 2 a b + b^{2}$ ${(a - b)}^{2} = a^{2} - 2 a b + b^{2}$ (a + b)² is not a² + b²
Difference of two squares	$a^{2} - b^{2} = (a + b) (a - b)$

Quadratic equations and graphs S2 · 5 G3

Zero Product Principle	$P Q = 0 \Rightarrow P = 0 or Q = 0$ only against a zero: rearrange to … = 0 first
Graph of y = ax² + bx + c	a > 0: opens upwards, minimum point · a < 0: opens downwards, maximum pointthe line of symmetry is halfway between the two x-intercepts

Algebraic fractions and formulae S2 · 6

Multiplying and dividing	$\frac{a}{b} \times \frac{c}{d} = \frac{a c}{b d}$ $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$ only the second fraction turns over
Turning a bracket round	$a - b = - (b - a)$

Direct and inverse proportion S2 · 7

Direct proportion	$y = k x$ $\frac{y_{1}}{x_{1}} = \frac{y_{2}}{x_{2}}$ quotients equal
Inverse proportion	$y = \frac{k}{x}$ $x_{1} y_{1} = x_{2} y_{2}$ products equal
Map scales	a length scale of 1 : n gives an area scale of 1 : n²1 km = 100 000 cm · 1 km² = 1 000 000 m² · 1 ha = 10 000 m²

Congruence and similarity S2 · 8

Similar figures	corresponding sides are in the same ratio — the scale factore.g. k = PQ ÷ AB for every pair of corresponding sides
Enlargement G3	$k = \frac{image length}{object length}$

Pythagoras' Theorem S2 · 9

Right-angled triangle, hypotenuse c	$c^{2} = a^{2} + b^{2}$ for a shorter side: a² = c² − b²
The converse	if c² = a² + b², the triangle is right-angled, with the right angle opposite c

Trigonometric ratios S2 · 10 G3

SOH · CAH · TOA

\sin A = \frac{opp}{hyp}

\cos A = \frac{adj}{hyp}

\tan A = \frac{opp}{adj}

for an acute angle A in a right-angled triangle

Volume and surface area of pyramids, cones and spheres S2 · 11

Volume of a pyramid	$\frac{1}{3} \times base area \times height$ not on the sheet, although the cone is
Total surface area of a cone	$π r l + π r^{2}$ the sheet gives only the curved part, πrl
Slant height	$l^{2} = r^{2} + h^{2}$ (cone) $l^{2} = h^{2} + {(\frac{a}{2})}^{2}$ (square pyramid)
Hemisphere	volume $\frac{2}{3} π r^{3}$ · curved surface $2 π r^{2}$ · solid total $3 π r^{2}$ half of the sphere’s formulae, plus the flat circle for a solid one

Probability of single events S2 · 12

Equally likely outcomes	$P(E) = \frac{number of favourable outcomes}{total number of possible outcomes}$
The two ends, and “not”	0 ≤ P(E) ≤ 1 · P(not E) = 1 − P(E)
Relative frequency	$\frac{number of occurrences}{total number of trials}$
Whole numbers from a to b inclusive	$b - a + 1$

Averages of statistical data S2 · 14

Mean of raw data	$\frac{sum of the data values}{number of data values}$ so sum = mean × number of values
Position of the median	$\frac{n + 1}{2}$ a position in the ordered list, not the median itself
Mode	the value (or values) with the highest frequency
Mid-value of a class	$\frac{lower boundary + upper boundary}{2}$ the x in Σfx ÷ Σf for grouped data — that formula is on the sheet

Curved surface area of a cone S2 · 11	$π r l$
Surface area of a sphere S2 · 11	$4 π r^{2}$
Volume of a cone S2 · 11	$\frac{1}{3} π r^{2} h$
Volume of a sphere S2 · 11	$\frac{4}{3} π r^{3}$
Area of triangle ABC Sec 3–4	$\frac{1}{2} a b \sin C$
Arc length Sec 3–4	$r θ$ θ in radians
Sector area Sec 3–4	$\frac{1}{2} r^{2} θ$ θ in radians

Sine Rule Sec 3–4	$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$
Cosine Rule Sec 3–4	$a^{2} = b^{2} + c^{2} - 2 b c \cos A$

Mean S2 · 14	$\frac{\sum f x}{\sum f}$
Standard deviation Sec 3–4	$\sqrt{\frac{\sum f x^{2}}{\sum f} - {(\frac{\sum f x}{\sum f})}^{2}}$