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GCSE/iGCSE Maths Formulae Sheet
Area of Parallelogram
Area of Rectangle
Area of Trapezoid
Circumference & Area:
Circle
Length of an arc
!
"
!
"
Cuboid Volume
Cylinder Surface Area
Cylinder Volume
Cone Surface Area
Cone Volume
Sphere Surface Area
Sphere Volume
Prism Volume
Pyramid Volume
Multiplication
Division
Negative Powers
Fractions
Find percentage of an
amount
Given % of an amount,
find the full amount
Increasing/decreasing
by a %
Density
base x height
!×#
#
$%&
#
$%&
Pressure
× 2()
Quadratic Function:
Solutions to
QRC + SR + T = U
Completing The Square
QRC ± SR + T = U
Max/Min Value
SSS
SAS
AAS
RHS
"##
&'(
"##
, c./ *° =
, 1!2 *° =
$%#
$%#
&'(
.
-
/
"
;0 = C √;E
/
-
Percentages
; as a percentage of :
;
× 100
:
Look for the words as a percent of
1233454674
852926:;
× 100
Look for the words percentage
gain/loss/increase/decrease
%
× ;HIJKL
100
MN9=K ;HIJKL
%
100
%
amountA1 ± B
Cumulative frequency
Box Plot
This is a running total of the frequencies
%
!±$%%
Interest: = ;HIJKL ×
%
× 1@UXY
5==
%
!&&
× LNH=
Note: Make sure t and % are same
unit of time
FV = PVA1 +
B
!&&
B
Fraction Of Amount
Improper to Mixed
Mixed to Improper
+ ;KZ − Fractions
× Fractions
÷ Fractions
Decimal to Fraction
Decimal to Percent
Fraction to Decimal
Fraction to Percent
Straight Line Equation
Methods to find
straight line equation
@
,(±√(&,E'F
"'
Finding a side:
!! = #! + % ! − 2#% cos <
Finding an angle:
# ! + % ! − !!
< = cos 45 =
>
2#%
1
!#/@2A
2
&
/
Finding a side:
=
=
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 ;
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
Geometry
• Slope intercept . = H- + %
• General ;- + :. + Z = 0
(get rid of fractions and move all on
one side to get into this form)
. = H- + %
Step 1: Find gradient H
• Given a graph – pick any 2 points on
BHIJ
the graph and use
Area of Triangle
Sine Rule
)*+,
)*+.
)*+0
&
/
=
=
)*+,
)*+.
)*+0
Finding an angle:
Circle Theorems
Angle at the centre is double the angle at the circumference
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°
Opposite angles of a cyclic quadrilateral add to DEC°
Watch out: All points need to be on the circumference!
Alternate segment theorem: The angle between a tangent and a side of a
triangle is equal to the opposite angle
For two intersecting chords, the products of their diagonals are equal
!# = %F
!(! + #) = %(% + F)
B?-
• Told parallel to another line⟹ same
slope
• Told perpendicular to another line
⟹”flip fraction and change the sign
(slopes multiply to make −1)
/ ,/
• Given 2 points. Use formula & $
FV=future value, PV=present value
L=times, )= interest rate
Quadratics
-=
&
/
=
=
)*+,
)*+.
)*+0
123 ,
123 .
123 0
Finding an angle:
=
=
&
/
Finding a side:
Fractions/Decimals/Percentages
Simplifying Fractions
Percent to Decimal
Percent to Fraction
=0>?-@
Non Right-Angled Trigonometry
Sine Rule
Cosine Rule
!&&
+ if increase
− if decrease
FV =amount + T!U.V21 ×
Exact Values
EF?GD?+/%
/C&)) H*'>$
/&>?I"F% EF?GD?+/%
Angle=
× 360
>">&C
$
+ if increase
− if decrease
Look for the words originally, at
the beginning, before…
Compound Interest
(interest added also
earns interest)
sin * ° =
Frequency density =
Pie chart
× () "
1 =Area of cross section x height
1
1 = × :;<= ;)=; × ℎ
3
Indices
- ' × - ( = - ')(
(- ' )( = - '(
(%- ' . ( )+ = % + - '+ . (+
-'
- ' ÷ - ( = ( = - ',(
1
- ,- = ,!
.
/
. .!
A B = ! and A B = !
Reverse percentage
Simple Interest
(interest on initial
amount)
SOHCAHTOA
Statistics
Frequency Density
3D Shapes
,+ = 2-. + 2-0 + 2.0
where x, y , z are side lengths
1 = -.0
where x, y , z are side lengths
,+ = 2()ℎ + 2() "
Note: Curved part: 2πrh
1 = () " ℎ
,+ = ()! + () "
Note: Curved part: ()!
where ! is slant length
1
1 = () " ℎ
3
,+ = 4() "
Note: Hemisphere 3() "
4
9 = () $
3
"
Note: Hemisphere= () $
/
Percentage gain/loss
(wants answer as a %)
speed=
% = 2(), + = () "
Fractional Powers
One amount as a % of
the other amount
(wants answer as a %)
x base x height
Pythagoras
'*)>&+/?
>*A?
A&))
density=
B"CDA?
E"F/?
pressure=
&F?&
Speed
(sum of parallel sides) x height
Area of a Sector
Cuboid Surface area
Right Angled Trigonometry
!! + # ! = % !
Hint: Given hyp⟹subtract, finding hyp⟹add
Compound Measures
2D Shapes
Area of Triangle
,; ≠0
Extra helpful facts to remember:
.&,.$
: "
:"
; W- ± X + % −
2;
4;
:"
%−
4;
Congruent Shapes
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
Proportion
. is … proportional to G
Directly: . = Y- , Inversely: . =
.
Straight Line Gradient
Distance between 2
points
(RK , \K ), (RC , \C )
Coordinates of
midpoint of (RK , \K ),
(RC , \C )
Circles
Step 2: Find . intercept %
• Given a graph – where the line
crosses the . intercept
• Plug point in(make sure this using
correct point-line must pass through
it).
.", .!
H=
-", -!
](-", -! )" + (.", .! )"
A
-!) -" .!) ."
,
B
2
2
(- − ;)" + (. − :)" = ) "
centre (;, :), radius )
Series (iGCSE only)
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
Geometric sequence:
u3 = ar 345
6(548!)
6(8!45)
S3 =
=
,r≠1
548
845
where ! =first term, r= common ratio
Differentiation (iGCSE only)
Rule
* + ⟹ 2* +45
Remember: Constants go to 0
'%
Turning/Stationary Points
Solve = 0
':
(Max/Min)
Proving whether
Use knowledge of shape of graph
Max/Min
+* ! happy face min
−* ! sad face max
;
+* max on left, min on right
;
−* min on left, max on right
Arithmetic sequence: