Volumes a)Triangle : Area = ½ base * height - universal Area of equilateral triangle = sqrt(3)/4 side 2 b)Rectangle : length * breadth c)Square : side 2 Diagonal = side * sqrt(2) Area = ½ product of diagonal d)Parallelogram : base * height e)Rhombus: ½ product of diagonals f)Circle : Area PiR 2 Circumference 2PiR g)Cuboid : is the rectangular solid having 6 faces with all the faces as rectangles Volumes : l*b*h Area for 4 walls : 2 (i+b)* h Total surface area of cuboid : 2 (lb + bh + lh) Body diagonals of cuboid : sqrt(l 2 +b 2 +h 2 ) h)Cube Volume = a 3 Total surface area of Cube 6 * a 2 i)Cylinder Volume Pi R 2 H Curved surface= 2PiRH Total surface = 2PiR(R+H) j)Cone Volume = 1/3 PiR 2 h Curved surface area = PiRL where L=Sqrt (R 2 +H 2 ) Total surface area : PiR(R+L) k)Sphere Volume : 4/3 PiR 3 Surface : 4 Pi R 2 Triangle GEOMETRY FORMULA
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Parallelogram : base * heighte) Rhombus: ½ product of diagonalsf) Circle : Area PiR2 Circumference 2PiRg) Cuboid : is the rectangular solid having 6 faces with all the faces as rectangles
a) Sum of angles is 180b) Exterior angle is equal to sum of interior angle non adjacent to it i.e. angles other
than the complementary angle of the exterior anglec) Sum of any two sides is more than the third sided) Equilateral triangle is the triangle with all the sides as same
Area = √ 3/4side2 Height √ 3/2side
Perimeter = 3Side
e) Right angle triangle
45 -90-45 triangle Hypotenuse = √ 2 * Side
30-60-90 triangle 30 side = ½ hypotenuse
60 side = ( √ 3/2) hypotenuse
f) If the angles of two triangles are same then they are similar then all the attributesthat they have will have same proportion – heights, sides etc.
Rectangle
a) Diagonals are equal and bisect each otherb) Diagonal = √ (a2+b2)
c) Of all the given rectangles of same area or perimeter square will have the maximumarea
Parallelogram
a) Diagonals bisect each other
b) Opposite angles are same
c) Each diagonal divides the parallelogram in triangles of same areaTrapezium
a) Only one pair of opposite side are parallel to each otherb) Area = ½ * (sum of parallel sides) * height
c) Isosceles trapezium is the one that is inscribed in a circle. The oblique sides areequal. The opposite angles made by oblique sides with the parallel side are equal.
Circle
a) Tangents drawn from an external side are equal
CUBOID: Let length=l, breadth=b and height=h units. Then,
Sum of N terms is given by Sn = a (r ⁿ - 1)/ ( r – 1)
Sum of infinite terms is given by S∞ = a/ ( 1 – r)
Where r is less than 1
Sum of infinite terms is given by S∞ = a/( r - 1)
Where r is greater than 1
Speed and Distance:
If a person is traveling from ‘A’ to ‘B’ with a km/ hr, and in return from
B to A with b km/hr his average speed is given by = 2 x a x b / (a+b) km/hr
If the distance traveled is same with two different speeds then average speed
is given by = 2 x a x b / (a+b)km/ hr
If the time taken is same with different speeds then average speed is given by = (a + b)/ 2 km/ hr. a km/ hr and b km/hr are different speeds.
Trains
If speed of the train is ‘A’ km/hr, length of the train is ‘B’ km. Then time taken to cross a pole in hrs = B/ A km/hr
If speed of the train is ‘A’ km/hr, length of the train is ‘B’ km.
Then time taken to cross a platform of length ‘C’ km in hrs =(B +C) km / A km/ hr .
Two trains ‘A’ and ‘B’ are traveling with a km/hr and b km/hrrespectively. Lengths of the train A and B are X km and Y km respectively.
Time taken to cross the slower train by faster train = (X +Y) km / (a+b) km/hr.
(If they are traveling in opposite direction)
Time taken to cross the slower train by faster train (If they are traveling in same
direction, Starting point of the faster train is at the end of the slower train)= (X +Y) km / │a - b│ km/hr.
Time taken to cross the slower train by faster train (If they are traveling in samedirection, Starting point of the faster train and slower train are on the same line)
If the value is increased successively by x% and y%,then final increase is
given by x+y+(xy/100) %
Conversions1 mile = 1609 meters
1 mile = 5280 feet.
1mile = 8 furlong1 furlong = 220 feet
1 km/hr = (5 / 18 ) m/s
Percentages
1. If the commodity price increases by x% then the consumption has to be reduced by
x/(100+x) to maintain the amount spent2. If the commodity price decreases by x% then the consumption has to be increased
by x/(100-x) to maintain the amount spent3. if a’s income is x% more than b on b’s income then b’s income is x/(100+x)% less
than a on a’s income4. if a’s income is x% less than b on b’s income then b’s income is x/(100-x)% more
than a on a’s income5. When it is stated that a is x% more than b then it means on the base of b. thus
a = b+x%b
Simple & compound interest
1. Simple interest = PNR/100 where p = principal; N=period (years) and R = rateAmount = P+I
2. compound interest = A = P (1+R/100)^NThis in compound interest the formula will give the final amount. Principal will have
to be deducted from it to get the interest portion. In case the interest is half yearly
then rate should be halfed and period should be doubled. Similarly for quarterly.Then the interest rate for different years is different then P (1+R1/100) (1+R2/100)
(1+R3/100)
3. population. The formula for compound interest can also be used for populations. Butwhere the populations is decreasing then the sign will change to ‘- ‘ instead of ‘+’
Averages
1.
Mode is the number that occurs most no of times in given sample
2. Median is the middle number of the given sample. Where the number of items ingiven sample is odd then (n+1)/2th number and if the number is even it is simpleaverage between n/2 and n+2/2th numbers
3. Arithmetic mean : it is sum of all numbers / no of numbers4. weighted average mean: it is sum of product of numbers and their respective
weights / sum of weights
5. Geometric mean: between two numbers it is x = Sqrt (ab). If geometric mean of onegroup of numbers (a) is X and that of group (b) is Y then geometric mean of boththe groups will be (X+Y)/ (a+b)
6. Harmonic mean : between two numbers = number of numbers / (sum of reciprocal of
numbers) ie. 2ab/(a+b). This also gives the average speed when same lengthdistances are covered in different speeds
7. GM^2 = AM*HM
Ratio proportion and variation
1. comparing two quantities as ratios:a. both the quantities should be of same kindb. both should have the same measurement per unitc. ratio is a pure number i.e. it does not have any measurement. It just denotes
how many times one quantity is of one of other2. compounding : if two different ratios (say a:b and c:d) are expressed in different
units then if we require to combine these two ratios then it will be AC:BD3. if a/b=c/d=e/f then the ratio is equal to a+c+e / b+d+f
Mixtures and allegation
1. Alligation ruleQuantity of cheap = Price of dear – average price
Quantity of dear Average price – price of cheap
2. if a vessel contains ‘a’ litres of liquid A and if ‘b’ litres are withdrawn and replaced byliquid B then if ‘b’ litres of the mixture is again withdrawn and replaced by liquid B.the operation is repeated ‘n’ number of times thenLiquid a left in vessel = ((a-b)/a)^n
Initial liquid in vessel
Profit or loss
If the a and b are two successive discounts that have been given then effective discount rate
1. while traveling if a person changes his speed in m:n ratio then the time taken willalso change in n:m ratio
2. if the A to B is traveled in T1 time and a speed and B to A if T2 time and b speedthen the average speed is give by(2ab) / (a+b) ………. Harmonic mean
Distance is given by (T1+T2) (2ab/(a+b))
Or (T1-T2) (2ab/(a-b))
Or (a-b) (T1T2 / (T1+T2))
3. if two persons start towards each other from different points and arrive at twopoints in a hrs and b hrs respectively after having met then ratio of their speed isgiven by SQRT (b) / SQRT (a) = a’s speed / b’s speed
Work
1. if A can do a work in x days then 1/xth work is done in one day2. if A is X times better workman than B then A will take 1/xth time of that taken by
B3. if A and B do work in X and Y days then they will complete the same work in XY /
(X+Y) days and in one day (X+Y)/ XY days work will be done4. if A and B can do a piece of work in X days and if A alone will be able to complete
the work in a days more than X and b can in b days more than X then X2 = ab5. if a pipe can fill a vessel in x hrs then 1/xth part of the vessel is filled in one hour6. if A pipe is X times bigger than B then A will take 1/X times lesser time than B7. if A and B fill the pipe in m and n hours respectively then both will fill the pipe in
MN / (m+n) hours and (m+n) / mn th part of vessel will be filled in one hour8. if one inlet pipe fills the vessels in M hrs and other pipe empties the vessel in N
hrs then the vessel will be filled in MN / (N-M) hrs. and (N-M)/MN the part will befilled
9. if an inlet pipes taken X minutes to fill the cistern and has taken a minutes longerthen the leak will empty the cistern in a*(1+a/x) minutes
10. A and B can fill the cistern in X hrs and A alone will fill the same in a minutesmore than X and b can fill it in b minuted more than X then X= sqrt (ab)
Clocks and Calendars
a) A dial of the clock is divided into 60 parts each called minute spaces
b) The hour hand goes 5 minute spaces in one hour and minutes hand goes 60 minutespaces in one hour. Thus the minute hand gains 55 minute spaces over the hourhand in one hour
c) When two hands are in 90 degree they are 15 minute spaces apart. This occurs twicein an hour.
d) When the two hands are in opposite directions they are 30 minute hands apart thisoccurs once in an hour
e) Two hands are in straight line when they coincide or are in opposite directionsf) The angle between the two hands = 6(x-11/12m)
X= hour hand convert into minute spaces i.e.* 5 of the earlier clock
M = the later part of the time i.e. minutes
g) The years that are divisible by 400 are the only ones that are leap year.
When mentioned RST then S will be the top vertices
Numbers
a) ODD +/- ODD = EVEN
b) ODD +/- EVEN = ODDc) EVEN +/- EVEN = EVENd) ODD * ODD = ODDe) ODD * EVEN = EVEN
f) EVEN * EVEN = EVENg) HCF of two numbers is the number that divides both the numbers exactlyh) LCM of two numbers is the number that is divided by both the numbers exactlyi) HCF*LCM= product of both the numbers
j) HCF of fraction is HCF of the numerators / LCM of denominatorsk) LCF of fractions is LCM of numerators and HCF of denominatorsl) if three numbers a,b,c are divided by N in such manner that r is the remainder each
time then smallest value of N is LCM of (a,b,c)+r
m) if three numbers a,b,c divide N is such manner that remainders are p,q,r then if (a-p)= (b-q) = (c-r) then the smallest value of N is LCM of (a,b,c) – (a-p)
Indices
a) Am * An=A(m+n) b) Am / An=A(m-n) c) (Am)n=A(m*n)
a) Arithematic progressionsSum = (n/2)*[2A+(n-1)d]
= (n/2) * (a+l)
Nth Term = A+(N-1)D
N = number of terms
D is the common differences
A is the first term
L is the last term
b) Geometric progression
Arn-1=Nth Term
Sum = A(1-rn)/(1-r)
Geometric mean = (ab)1/2
c) Harmonic mean = it is the arithmetic mean of reciprocals of numbersSum and nth number of harmonic mean is reciprocal of arithmetic mean
Harmonic mean of two numbers is 2ab/(a+b)
Permutation and combination
a) Fundamental principal of addition: if one thing can be done m number of ways andother thing can be done in n number of ways independent of other. Then either ofthem can be done in (m+n) ways
b) Fundamental principal of multiplication: if one thing can be done m number of ways
and other thing can be done in n number of ways independent of other. Then eitherof them can be done in (m*n) ways
c) Permutation : permutation of n objects taken r at a time is the arrangement in a
straight line of r objects taken at a time denoted by N!/(N-R)!d) The number of permutation of n objects taken all at a time = n!e) The number of permutations of n objects taken all at a time when p of them are like,
q are like = n!/p!q!
f) Combination is the selection of r objects in n objects. Denoted as N!/(n-r)!r!
g) Number of permutations of n objects taken all at a time in circle (n-1)!h) When the repetition of allowed then permutation nr