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GCSE/iGCSE Maths Formulae Sheet
!
!
"
Circumference & Area:
Circle
Length of an arc
speed=
Density
base x height
!×#
Pressure
!! + # ! = % !
if given hyp⟹subtract, if finding hyp⟹add
Pythagoras
'*)J&+/F
J*HF
H&))
density=
E"QIHF
R"N/F
pressure=
&NF&
Speed
x base x height
"
Area of Parallelogram
Area of Rectangle
Area of Trapezoid
Right Angled Trigonometry
Compound Measures
2D Shapes
Area of Triangle
SOHCAHTOA
sin * ° =
"##
$%#
, c./ *° =
&'(
,
$%#
1!2 *° =
"##
&'(
(sum of parallel sides) x height
% = 2(), + = () "
Exact Trig Values
#
$%&
#
Area of a Sector
× 2()
$%&
× () "
3D Shapes
!" = 2%& + 2%( + 2&(
where x, y , z are side lengths
Cuboid Surface area
Frequency Density
) = %&(
where x, y , z are side lengths
!" = 2*+ℎ + 2*+ !
Note: Curved part: 2πrh
) = *+ ! ℎ
!" = *+1 + *+ !
Note: Curved part: *+1, 1 is slant length
1
) = *+ ! ℎ
3
!" = 4*+ !
Note: Hemisphere 3*+ !
4
5 = *+ <
3
!
Note: Hemisphere= *+ <
Cuboid Volume
Cylinder Surface Area
Cylinder Volume
Cone Surface Area
Cone Volume
Sphere Surface Area
Sphere Volume
<
Pie chart
Angle=
Cumulative frequency
Box Plot
<
Negative Powers
Fractions
/
.
-
Fractional Powers
/
$
40 = 5 √47
Fraction Of Amount
Improper to Mixed
.#
/#
-
Mixed to Improper
Percentages
8 as a percentage of 7
8
× 100
7
Look for the words as a percent of
One amount as a % of
the other amount
(wants answer as a %)
=2>>?9?3@?
Percentage gain/loss
(wants answer as a %)
A92B237C
× 100
Look for the words percentage
gain/loss/increase/decrease
%
× 8=>?@A
100
Find percentage of an
amount
Given % of an amount,
find the full amount
Increasing/decreasing
by a %
D*EF+ &H"I+J
%
&''
× Fractions
÷ Fractions
Decimal to Percent
Fraction to Decimal
,H"I+J
Reverse percentage
+ 8@P − Fractions
Decimal to Fraction
%
amountB1 ± D
5LL
+ if increase
− if decrease
%
5±&''
+ if increase
− if decrease
Look for the words originally, at the
beginning, before…
Pyramid Method For Percentages:
Cosine Rule
Finding a side:
Area of Triangle
Finding an angle: < = cos 45 =
!-/
1
!#/@2A
2
&
Finding a side:
=
=
Sine Rule
-!6/ !4&!
)*+,
&
)*+,
Finding an angle:
)*+.
)*+.
=
>
/
)*+0
/
)*+0
=
Fractions/Decimals/Percentages
Simplifying Fractions
Indices
Division
× 360
Finding an angle:
Circle Theorems
5
, ' × , ( = , ')(
(, ' )( = , '(
(%, ' / ( )+ = % + , '+ / (+
,'
, ' ÷ , ( = ( = , ',(
,
1
, ,- = ,
. ,/#
. 2 3 = # and 2 3 =
/&JFD"N% RNFSIF+/%
J"J&Q
This is a running total of the frequencies
&
/
=
=
)*+,
)*+.
)*+0
123 ,
123 .
123 0
=
=
&
/
!! = #! + % ! − 2#% cos <
Finding a side:
RNFSIF+/%
/Q&)) T*'J$
Angle at the centre is double the angle at the circumference
) = × 789: 8+:8 × ℎ
Multiplication
Sine Rule
Frequency density =
) =Area of cross section x height
Prism Volume
Pyramid Volume
Non Right-Angled Trigonometry
Statistics
Fraction to Percent
Percent to Decimal
Percent to Fraction
Step 1: Find a factor of both numbers i.e. a
number that fits in both the numerator AND
denominator
Step 2: Say how many times for each
Step 3: Check whether you can do steps 1
and 2 again.
&
of amount
Step 1: Divide amount by b
Step 2: Multiply answer found by 8
Step 1: Divide the numerator by the
denominator.
Step 2: Write down the whole number a
Step 3: Put the remainder in the numerator.
The new denominator remains the same as
that of the original improper fraction.
Step 1: Multiply the whole number by the
fraction’s denominator
Step 2: Add that to the numerator
Step 3: Then write the result on the top of
the original denominator
Need a common denominator (the smallest
number that that both the numerator and
denominator fit into)
Don’t need common denominator.
Can cancel diagonally or vertically, not
horizontally.
Don’t need common denominator.
“Keep change flip”
Write over 10,100,1000 etc depending on
how many places after the decimal and
simplify.
Multiply by 100
Write as an equivalent fraction over
10,100,1000 etc and then easy to divide by
this number
OR
Use short division if can’t write as an
equivalent fraction
Turn into a decimal and then just a decimal
to percent question i.e. multiply decimal
found by 100
Divide by 100
Write over 100 and simplify
Angles subtended in the same
segment by a chord are equal
Angle in a semicircle is a right
angle
A tangent meets a radius at BC°
Alternate segment theorem: The
angle between a tangent and a
side of a triangle is equal to the
opposite angle
Opposite angles of a cyclic
quadrilateral add to DEC°
Watch out: All points need to be
on the circumference!
Tangents which meet at a point
are equal in length
For two intersecting chords, the products of their diagonals are equal
!# = %F
!(! + #) = %(% + F)
Extra helpful fact to remember
Geometry
Simple Interest
(interest on initial
amount)
amount + Bamount ×
%
5LL
%
× timeD
Interest: = amount ×
× time
5LL
Note: Make sure t and % are same unit
of time
Compound Interest
(interest added also
earns interest)
N
J
FV = PVB1 +
D
5LL
FV=future value, PV=present value
A=times, += interest rate
Straight Line Equation
Methods to find
straight line equation
Quadratic Function:
Solutions to
891 + ;9 + < = =
Completing The Square
891 ± ;9 + < = =
Max/Min Value
%=
8 B% ±
!&
-
!
,8 ≠0
D +O−
!&
O−
•
-(
•
P&
-(
P&
Congruent Shapes
SSS
SAS
AAS
RHS
Three sides of each triangle equal
Two sides and included angle equal
Two angles and corresponding side
equal
Contains right angle and hypotenuse
and another side equal
Direct/Indirect Proportion
/ is … proportional to ,
2
Directly: / = ?, , Inversely: / =
.
Slope intercept & = =% + O
General 8% + 7& + P = 0
(get rid of fractions and move all on one
side to get into this form)
& = =% + O
Step 1: Find gradient =
• Given a graph – pick any 2 points on the
N*)F
graph and use
•
Quadratics
4-±O-(4P&/
•
•
Straight Line Gradient
Distance between 2
points (WU , XU ), (WV , XV )
Coordinates of midpt of
(WU , XU ), (WV , XV )
Circles
NI+
Told parallel to another line⟹ same
slope
Told perpendicular to another line
⟹”flip fraction and change the sign
(slopes multiply to make −1)
% 4%
Given 2 points. Use formula ( &
Functions
Inverse
Replace !(#) with %, swap # & %, solve for %
Composite
!((#) means plug ((#) into !(#)
Transformations
a=vertical stretch sf ), b=horizontal stretch sf "
c=translation c units x direction, d=translation d
units in y direction
!(−#)=reflcn in %, axis − !(#)=reflcn in # axis
!
)!(*# + ,) + “anything in a bracket affects
# and does the opposite”
Series (iGCSE only)
Arithmetic sequence:
IJK term:
u3 = a + (n − 1)d
sum of n terms
n
2
S3 = [2a + (n − 1)d] = (! + P)
2
2
where ! =first term, d= common diff,
P=last term
u3 = ar 345
7(549")
7(9"45)
S3 =
=
,r≠1
549
945
where ! =first term, r= common ratio
;(4;&
Step 2: Find & intercept O
• Given a graph – where the line crosses
the & intercept
• Plug point in and solve for c (make sure
this is correct point-line passes through).
&!4 &5
==
%!4 %5
T(%!4 %5 )! + (&!4 &5 )!
B
;&);( %&)%(
!
,
!
D
(% − 8)! + (& − 7)! = + !
centre (8, 7), radius +
Geometric sequence:
Differentiation (iGCSE only)
Rule
Turning/Stationary Points
(Max/Min)
Proving whether
Max/Min
* + ⟹ 2* +45
Remember: Constants go to 0
Solve
'%
';
=0
Use knowledge of shape of graph
+* ! happy face min
−* ! sad face max
<
+* max on left, min on right
<
−* min on left, max on right