Download Higher Tier Formulas

Dyffryn School
Ysgol Y Dyffryn
Mathematics Faculty
Formulae and Facts
Booklet
Higher Tier
Number Facts
Sum
Difference
Product
Share
Prime number
Square number
Cube number
Factor
Common factor
Highest Common Factor
Multiple
Common Multiple
Lowest Common Multiple
This means add.
This means take away.
This means multiply.
This means divide.
A number that has only two factors and can only be divided by
itself and one. (1, 9, 21, 27 are not prime numbers.)
The answer to when a number is multiplied by itself.
The answer to when number is multiplied by itself 3 times.
A number that can divide into another number e.g. the factors of 6 are 1,
2, 3 and 6
A number that can divide into two or more other numbers
The largest common factor.
A number that another number can divide into e.g. Multiples of 10 are 10,
20, 30 …
A number that two or more numbers can divide into
The smallest common multiple
Averages
Mean – Total divided by the amount of numbers.
Median – Middle number of an arranged set.
Mode – The most common number.
Range – The difference between the largest and smallest numbers.
Area
Square = (length)2
Rectangle = Length × Width
Triangle = Base × Height
2
Parallelogram = Base × Height
Trapezium = (a + b) × height
2
Compound/composite shapes – Split these up into known shapes and add up the areas.
Surface area of a cone = rl
Surface area of a sphere = 4r2
Circle
Diameter is the distance across the centre of a circle
Radius = ½ diameter
Area of a circle =  × radius2
Circumference of a circle =  × diameter
For a semi-circle find the area as normal then half the answer
Area of sector =

× r2
360
Length of an arc =

360
× D
Volume
Volume of any prism = cross sectional area (Face area) × length
Cube = (length)3
Cuboid = length × width × height
Cylinder =  r2 h
Cone = 1/3  r2 h
Sphere = 4/3  r3
Square based pyramid = 1/3 × area of the base × height
Density
Density = Mass
Volume
M
D
Average Speed
Average speed = Distance
Time
V
D
S
T
Right angled triangles
Pythagoras’ Theorem
c2 = a2 + b2
or a2 + b2 = c2 (The hypotenuse is the longest side and is opposite the right angle)
Trigonometry
SOH CAH TOA
Non-right angled Triangles
Finding an angle
Cosine rule
Cosine rule to find an angle
Sine rule
Sine rule to find an angle
Finding a side
Cosine rule
Must have all three sides.
Cos A = b2 + c2 – a2
2bc
Must know a side and an angle as well as the side opposite the
angle you are trying to find.
Sine A = Sine B = Sine C
a
b
c
Need to know two sides and the angle between the 2 sides. You are
trying to find the third side.
Cosine rule
a2 = b2 + c2 – 2bc cosA
Sine rule
Need to know a side and an angle as well as the angle opposite the side
you are trying to find. (look for pairs)
Sine rule
a
= b
=
c
Sine A Sine B
Sine C
Area of a triangle
½ absinC
Foreign Exchange
To convert from the original money to new money multiply
To convert back divide
Scale drawing
To go from scale to real life multiply
To go from real life to scale divide
Bearings
Bearings are always measured from the North and are measured in a clockwise direction. If a bearing
is bigger than 180o, measure anticlockwise and subtract from 360.
Estimated Mean
Estimated mean = (Total frequency x midpoint)  Total frequency
Grouped frequency diagram
Frequency polygon
or
total f × x
Total f
Bar chart with a proper scale
Plot across to the midpoint and up to frequency. Join the points
plotted in order with a ruler.
Units (You have to know these)
1cm = 10mm
100cm = 1m
1000m = 1km
1g = 1000mg
1kg = 1000g
1 litre = 1000cm3
1 litre = 1000 millilitres
1 litre = 100 centilitres
1 mile  1.6 km
1m3 = 1000 litres
1ml = 1cm3
1 gallon = 4.5 litres
1 inch  2.5 cm
Converting squared or cubed units
cm2
mm2 × 102 or 100
2
m
cm2 ×1002 or 10000
3
m
cm3
× 1003 or 1000000
2
2
mm
cm
 102 or 100
cm2
m2
 1002 or 10000
3
3
cm
m
 1003 or 1000000
Metric Units
Weight
Grams (g)
Kilograms (kg)
Tonnes
Length
Millimetres (mm)
Centimetres (cm)
Metres (m)
Kilometres (km)
Volume
Millilitres (ml)
Centilitres (cl)
Litres (l)
Grams are used for the weight of small items, for example the
weight of a pound coin or the weight of an exercise book.
Kilograms are used for the weight of heavy items,
for example, the weight of a sofa or the weight of a person.
Tonnes are used for the weight of very heavy items, for example
The weight of an aeroplane.
Millimetres are used for very small lengths, for example
the width of a finger nail.
Centimetres are used for small lengths, for example
the length of an exercise book.
Metres are used for distances such as the length of a room,
the height of a person or the height of a door.
Kilometres are used for long lengths such as the distance
between Swansea and Cardiff.
Millilitres are used for small volumes, for example the volume
water in a teaspoon.
Centilitres can be used to measure liquid in a glass.
Litres are used for larger volumes, for example the volume of
petrol in a tank.
Angle Facts
Angles on a straight line add up to 180o
Angles round a point add up to 360o
Angles in a triangle add up to 180o
Angles in a quadrilateral add up to 360o
Vertically opposite angles are equal
Corresponding angles (F) formed in parallel lines are the same
Alternate angles (Z) formed in parallel lines are the same
Interior angles (C) formed in parallel lines add to 180
Circle Theorems
Angle at centre twice one at circumference
Angles subtended on the same arc are equal
Opposite angles in a cyclic quadrilateral add to 180
Angle in semicircle is 90
The radius meets a tangent at 90
The angle between a chord and tangent creates an equal angle in the alternate segment
Two tangents from one point outside the circle are equal in length
A line drawn from the centre at 90 to a chord bisects the chord.
Polygons
The angle facts here are for REGULAR POLYGONS
Polygon
No of sides
Sum of angles Interior angle
Triangle
3
180o
60o
o
Quadrilateral
4
360
90o
o
Pentagon
5
540
108o
o
Hexagon
6
720
120o
Exterior angle
120o
90o
72o
60o
The sum of the interior angles can be found by using (n – 2) × 180. To then find each angle in a
regular polygon divide what the sum of the interior angles is by the number of sides.
The sum of the exterior angles in any polygon is 360o
Shape Facts
Triangle
Types
Equilateral
Isosceles
Right angle
Scalene
All sides are the same length, all angles are 60o
Two sides the same length, two angles the same
Triangle with a right angle in it
All sides have different lengths
Quadrilateral
Types
Square
Rectangle
Rhombus
All sides the same length, all angles are 90o
Opposite sides are the same length, all angles are 90o
All sides are the same length, opposite sides are parallel, opposite angles are
the same
Parallelogram Opposite sides are parallel and equal in length, opposite angles are the same
Trapezium
One set of parallel sides
Kite
Two sets of equal sides, two horizontal angles are the same
3D Shapes
Cube
Cuboid
Triangular prism
Square based pyramid
Tetrahedron
All faces are congruent, this means the same (square)
Either all faces are rectangles or 4 are rectangles and 2 are squares
2 faces are triangles and 3 are rectangles
5 vertices (corners), 4 faces are triangles and 1 is a square
All faces are equilateral triangles.
Dimensions
1 dimension = Length
2 dimensions = Area
3 dimensions = Volume
Factorising
12a + 3
use one bracket, common factors
3x2 + 2x
use one bracket, common factors
4a2b – 16b
use one bracket, common factors
x2 + 7x + 10 use two brackets, quadratic
9x2 – 16
use two brackets, difference of two squares
2
3x – 5x – 9
use two brackets, harder quadratics
Quadratic formula
x = -b +
b2 – 4ac
-- 2a
Compound interest
Use 1st year, 2nd year, 3rd year etc. Remember to add on the interest found each year to get a new
total/balance for each year. The compound interest is the end total – what you started with.
[or A = P(1 + R/100)n]
Depreciation
This is the same as compound interest but instead of adding the interest each year you subtract it.
[ or A = P(1 - R/100)n]
Percentage Change
% change = change/original × 100
Histograms
Frequency Density = frequency  class width
Frequency = frequency density × class width
or
Standard Form
This is
(a number between 1 and 10) × 10power
The power is positive if the given number is > 1.
The power is negative if the given number is > 0 but < 1.
find the area of each bar