MATH FORMULAS

On this page you can find many math formulas in different topics of math.It is more useful to the students in all grades. Once you memorize this kind of formulas you can solve any difficult in an easy way.We have covered all most all the basic topics in math. After memorizing this formula you can try our worksheets and quiz. We have also given shortcuts to memorize these formulas. In each category you can get many formulas and also shortcut ideas.In each category we have example problems.


          Topics

                Math Formulas


Addition formula

ab + cd = (ad + bc) bd

Subtraction formula

ab - cd = (ad - bc) bd

Multiplication formula

ab x cd = (a x c) (b x d)

Division formula

ab / cd = (a x d) (b x c)




          Algebra

                Math Formulas


Formula for square

(a + b)² = a² + 2 ab + b²

(a - b)² = a² - 2 ab + b²

a² - b² = (a + b) (a - b)

a² + b² = (a + b)² - 2ab

a² + b² = (a - b)² + 2ab

a² + b² = ½ [(a + b)² - (a - b)²]

ab = ¼ [(a + b)² - (a - b)²]

(a - b)² = (a + b)² - 4 ab

(a + b)² = (a - b)² + 4 ab

(a+b+c)² =a²+b²+c²+2ab+2bc+2ca

(a-b+c)² =a²+b²+c²-2ab-2bc+2ca

(a-b-c)² =a²+b²+c²-2ab+2bc-2ca

Formula for cube

(a + b)³ = a³ + 3 a² b  + 3 a b² + b³

(a - b)³ = a³ - 3 a² b  + 3 a b² - b³

a³ + b³ = (a + b)³ - 3 ab (a + b)

a³ - b³ = (a - b)³ + 3 ab (a - b)

a³ + b³ = (a + b) (a² - ab + b²)

a³ - b³ = (a - b) (a² + ab + b²)

(x + a) (x + b) = x² + (a + b) x + ab

(x + a) (x + b) (x + c) =

x³+(a+b+c) x²+(ab+bc+ca) x+abc

a² + b² + c² - a b - b c - c a =

½[(a-b)²+(b-c)²+(c-a)²]

(a+b+c) (a² + b² + c² - a b-b c-c a) =

a³ + b³ + c³ - 3 a b c



          Logarithms

                Math Formulas


Product rule

log ₐ (m n) =  log ₐ m + log ₐ n

Quotient rule

log ₐ (m/n) =  log ₐ m - log ₐ n

Power rule

log ₐ m ⁿ = n log ₐ m

Change of base rule

log ᵤ m = (logᵥm) x (logᵤv)

Reciprocal rule

log ₐ i = 1/ logᵢ a

log ₐ 1 = 0

log ₐ a = 1




          Exponents

                Math Formulas


Product rule

a m x a n = a (m + n)

Quotient rule

am⁄an = a (m-n)

Power rule

(am)n = amn

Combination Law

(am x bm) = (axb)m
1 ⁄a m = a-m
a 0 = 1




Set theory

Math Formulas


Associative law

      A ∪ (B∪C)=(A∪B) U C

      A ∩ (B∩C)=(A∩B) ∩ C

Distributive law

    A ∪ (B∩C) = (A∪B) ∩ (A∪C)

    A ∩ (B∪C) = (A∩B) ∪ (A∩C)

De Morgan's law

      (i)(A∪B)'=A'∩B'.

      (ii)(A∩B)'=A'∪B'.

      (iii)A-(B∪C)=(A-B)∩(A-C)

      (iv)A-(B∩C)=(A-B)∪(A-C)

Cardinal number of power set

      n [p(A)] = 2 ⁿ

Identity laws

      A∪∅=A

      A∩U=U

Domination laws

      A∪U=U

      A∩∅=∅

Idempotent laws

      A∪A=A

      A∩A=A

Commutative laws

      A∪B=B∪A

      A∩B=B∩A




Sequence and series

Math Formulas


General form of an arithmetic progression 

a,(a+d),(a+2d),(a+3d),...........

a - first term d = common difference (t₂ - t₁)

nth term or general term of an A.P

tn = a + (n - 1)d

Sum of n terms of an A.P

sn = n2 [2a + (n-1)d]
sn = n2 [a + L]

L-last term

number of terms of an A.P

n = (l-a)d + 1

General form of g.p

a,ar,ar2,ar3,.................

a - first term r = common ratio (t₂/t₁)

Sum of n terms of g.p 

if r≠1 sn = a( 1- rn)⁄( 1 - r)
sn = a( rn - 1 )⁄( r - 1)

if r = 1 sn = n a
Sum of infinite series sn = a ⁄( 1 - r)




Trigonometry

Math Formulas


Trigonometric ratios

sin θ = Opposite side/Hypotenuse side

cos θ = Adjacent side/Hypotenuse side

tan θ = Opposite side/Adjacent side

Cosec θ = Hypotenuse side/Opposite side

Sec θ = Hypotenuse side/Adjacent side

cot θ = Adjacent side /Opposite side

Reciprocal

sin θ = 1/Cosec θ

Cosec θ = 1/sin θ

Cos θ = 1/sec θ

sec θ = 1/cos θ

tan θ = 1/cot θ

cot θ = 1/tan θ

identities

sin² θ  + cos² θ = 1

sin² θ  = 1 - cos² θ

cos² θ = 1 - sin² θ

Sec² θ - tan² θ = 1

Sec² θ  = 1 +  tan² θ

tan² θ  =  Sec² θ - 1

Cosec² θ - cot² θ = 1

Cosec² θ = 1 + cot² θ

cot² θ =  Cosec² θ - 1

Complementary angles

Sin (90 - θ) = cos θ

Cos (90 - θ) = sin θ

Tan (90 - θ) = cot θ

Cot (90 - θ) = tan θ

Cosec (90 - θ) = sec θ

Sec (90 - θ) = cosec θ 

Values of certain angles

Double angle formula

Sin 2A = 2 Sin A cos A

Cos 2A = cos² A - Sin² A

tan 2A = 2 tan A/(1-tan² A)

Cos 2A = 1 - 2Sin² A

Cos 2A = 2Cos² A - 1

sin 2A = 2 tan A/(1+tan² A)

cos 2A = (1-tan² A)/(1+tan² A)

sin²A = (1-Cos 2A)/2

Cos²A = (1+Cos 2A)/2

Half angle formula

Sin A = 2 Sin (A/2) cos (A/2)

Cos A = cos² (A/2) - Sin² (A/2)

 tan A = 2 tan (A/2)/[1-tan² (A/2)]

Cos A = 1 - 2Sin² (A/2)

Cos A = 2Cos² (A/2) - 1

 sin A = 2 tan (A/2)/[1+tan² (A/2)]

cos A = [1-tan²(A/2)]/[1+tan² (A/2)]

 sin²A/2 = (1-Cos A)/2

Cos²A/2 = (1+Cos A)/2

 tan²(A/2) = (1-Cos A)/(1+Cos A)


Compound angle formula

Sin (A+B) = Sin A Cos B + Cos A Sin B

Sin (A-B) = Sin A Cos B - Cos A Sin B

Cos (A+B) = Cos A Cos B - Sin A Sin B

Cos (A-B) = Cos A Cos B + Sin A Sin B

tan (A+B) = [Tan A + Tan B] /(1- Tan A Tan B)

tan (A-B) = [Tan A - Tan B] /(1+ Tan A Tan B)

Compound angles sum and differences

sin(A+B)+Sin (A-B) = 2 Sin A cos B

sin(A+B)-Sin (A-B) = 2 Cos A sin B

cos(A+B)+Cos (A-B) = 2 Cos A cos B

cos(A+B)-Cos (A-B) = -2sin A sin B

3A formula

Sin 3A = 3 Sin A - 4 sin³A

Cos 3A = 4 Cos³A - 3 Cos A

tan 3A=(3 tan A - tan³A)/(1-3tan²A)


Sum to product formula

sin C + sin D = 2 sin [(C+D)/2] cos [(C-D)/2]

sin C - sin D = 2 cos [(C+D)/2] sin [(C-D)/2]

cos C + cos D = 2 cos [(C+D)/2] cos [(C-D)/2]

cos C - cos D = -2 sin [(C+D)/2] sin [(C-D)/2]




Analytical geometry

Math Formulas


Section formula internally

Section formula externally

Area of triangle using

12 {x1(y2-y3) + x2(y3-y1) + x3(y1-y2)}

Area of quadrilateral 

12{(x1y2+x2y3+x3y4+x4y1)-(x2y1+x3y2+x4y3+x1y4)}

Centroid

(x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3

Midpoint of the line segment

(x₁ + x₂)/2 , (y₁ + y₂)/2

Distance between two points

√(x₂ - x₁) ² + (y₂ - y₁) ²

Slope of the line

m = tan θ

m = (y2 - y1)/(x2 - x1)

m = - coefficient of x / coefficient of y

y = m x + b

m-slope

Equation of the line

Slope intercept form:

y = m x + b

Here m = slope and b = y-intercept

Two point form:

(y-y)/(y₂-y₁) = (x-x₁)/(x₂-x₁)

Point- Slope form:

(y-y1) = m (x-x1)

Intercept form:

(X/a) + (Y/b) = 1

Perpendicular distance a point and a line

d=| (ax₁+by₁+c)/ va²+b²|

Angle between two lines

θ = tan-¹ |(m₁ - m₂)/(1 + m₁ m₂)|

Equation of circle

(x-h)² + (y-k)² = r² 

Equation of circle with two endpoints of diameter

(x-x₁) (x-x₂) + (y-y₁) (y-y₂) = 0

General equation of circle

x² + y² + 2gx + 2fy + c = 0

Length of the tangent

√ (x₁² + y₁² + 2gx₁ +2fy₁+c)

Condition of two circles touching externally

c₁c= r₁ + r₂

Condition of two circles touching internally

 C₁ C = r₁ - r₂

Orthogonal circles

2 g₁g₂+2f₁f₂=c₁+c₂




Differentiation

Math Formulas


d(xⁿ)n xⁿ⁻¹

d(sin x) = cos x

d(cos x)  - sin x

d(logₐ x) = 1/x log ₐ e

d(logₑ x) = 1/x

d(tan x) = sec² x

d(sec x)= sec x tan x

d(cot x)= - cosec²x

d(cosec x)= - cosec x cot x

d(Sin -¹ x)=  1/√(1-x²)

d(Cos -¹ x) = -1/√(1-x²)

d(tan -¹ x)= 1/(1+x²)

d(Cot -¹ x)= -1/(1+x²)

d(Sec -¹ x)=  1/[x√(x² - 1)]

d(cosec -¹ x)= -1/[x√(x² - 1)]

Product rule

(UV)' = UV' + VU'

Quotient rule

(U/V)' =  [VU' - UV'] /V²

Lagrange theorem

1.f(x) is defined and continuous on the closed interval [a,b] 

2.f(x) is differentiable on the open interval (a,b).

Then there exists at least one point c ∊ (a,b) such that 

f ' (c) = f (b) - f (a) / (b - a)

Rolle's theorem

1.f(x) is defined and continuous on the closed interval [a,b] 


2.f(x) is differentiable on the open interval (a,b).

3.f(A) = f(b)


then there exists at least one point c ∊ (a,b) such that f ' (c) = 0

Maclaurin series

f(x) = f(0) + (f'(0) ⁄1!)x + (f''(0) ⁄2!)x2 + (f'''(0) ⁄3!)x3 + ..........

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Integration

Math Formulas


∫ xⁿ dx = x⁽ⁿ ⁺ ¹⁾/(n + 1) + c

∫ (1/xⁿ) dx = -1/(n - 1) x⁽ⁿ⁻¹⁾  + c

∫ (1/x) dx = log x  + c

∫ e^(x) dx = e^x  + c

∫ a^(x) dx = a^x/(log a)  + c

∫ Sin x dx = - Cos x + c

∫ Cos x dx = Sin x + c

∫ Cosec ² x dx = - Cot x  + c

∫ Sec ² x dx = tan x  + c

∫ sec x tan x dx = sec x  + c

∫ Cosec x cot x dx = - Cosec x  + c

∫ 1/(1 + x ²) dx =  tan ⁻ ¹x  + c

∫ 1/ √(1 - x ²) dx = Sin ⁻ ¹x   + c

∫ (ax + b)ⁿ dx = (1/a) (ax + b)⁽ⁿ ⁺ ¹⁾/(n + 1)   + c

∫ 1/(ax + b) dx = (1/a) log (ax + b) + c

∫ e^(ax + b) dx = (1/a) e^ (ax + b) + c

∫ Sin (ax + b) dx = -(1/a) Cos (ax + b) + c

∫ Cos (ax + b) dx = (1/a) Sin (ax + b) + c

∫ Sec ² (ax + b) dx = (1/a) tan (ax + b) + c

∫ Cosec ² (ax + b) dx = -(1/a) cot (ax + b) + c

∫ Cosec (ax+b)cot (ax+b)dx=-(1/a)Cosec (ax+b) + c

∫ sec (ax + b) tan (ax + b) dx = sec (ax + b) + c

∫ 1/1+ (ax) ² dx = (1/a) tan ⁻ ¹ (ax) + c

∫ 1/ √[1 - (ax ²)] dx = (1/a) Sin ⁻ ¹(ax)   + c

Integrating by parts

∫ u dv  = uv - ∫ v du




Mensuration

Formulas


Area of circle

Area of circle = Π r ²

Circumference of circle = 2 Π r

Area of triangle

Area of Equilateral-triangle = (√3/4) a²

Perimeter of Equilateral-triangle = 3a

Area of scalene triangle

Area of scalene triangle = √s(s-a)(s-b)(s-c)

Perimeter of scalene triangle

= a + b +  c

Area of semicircle

Area of Semi circle= (1/2) Π r²

Perimeter of semi-circle = Πr

Area of quadrant

Area of quadrant = (1/4) Π r²

Area of rectangle

Area of rectangle = L x W

Perimeter of rectangle=2(l+w)

Area of square

Area of square = a²

Perimeter of square = 4a

Area of parallelogram

Area of parallelogram = b x h

Area of quadrilateral

Area of quadrilateral

=(1/2) x d x (h₁+h₂)

Area of rhombus

Area of rhombus =(1/2) x (d₁ x d₂)

Area of trapezoid

Area of trapezoid =(1/2) (a + b) x h

Area of sector

Area of the sector = (θ/360) x Π r ² square units

(or)  Area of the sector = (1/2) x l r square units   

Length of arc = (θ/360) x 2Πr

Cylinder

Curved surface area = 2 Π r h

Total Curved surface area = 2 Π r (h+r)

Volume = Π r²h

Cone

Curved surface area = Π r l

Total Curved surface area = Π r (L+r)

Volume = (1/3)Π r²h

L² = r² +

sphere

Curved surface area = 4Π r²


Volume = (4/3)Π r³

Hemisphere

Curved surface area = 2Π r²

Total Curved surface area=3Π r²

Volume = (2/3)Π r³

Cuboid

Curved surface area=4h(l+b)

Total surface area = 2(lb+bh+h l)

Volume = l x b x h

Cube

Curved surface area=4a²

Total surface area = 6a²

Volume = a³







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