Square of Numbers

Percentage 

CUBE of Numbers

ALGEBRA 

• (a – b)² = a² + b² – 2ab  = (a+ b)² – 4ab
• a² +  b²  =  (a – b)² + 2ab
• (a + b)² = a² + b² + 2ab = (a – b)² + 4 ab
• a² + b²  =  (a + b)² – 2ab
• (a + b)² + (a – b)² = 2(a² + b²)
• (a + b)²  – (a – b)² = 4ab
• a² – b² =  (a – b)(a + b)
• (a + b + c )² =  a² + b² + c² + ( 2ab + 2bc +2ca )

• (a + b)³ = a³ + b³+ 3ab(a+ b) = a³ + b³+ 3a²b + 3ab²
• a³ + b³ = (a + b)³ – 3ab (a+ b)
• (a – b)³ = a³ – b³– 3ab(a – b) = a³ – b³– 3a²b + 3ab²
• a³ – b³ = (a – b)³ + 3ab (a- b)
• (a + b)³ + (a – b)³ = 2a(a² + 3b²)
• (a + b)³ – (a – b)³ =2b(b² + 3a²)
• a³ – b³ = (a – b)(a² + b² + ab ) = (a – b)³ + 3ab(a – b)
• a³ + b³ = (a + b)(a² + b² – ab ) = (a + b)³ – 3ab(a + b)
•  a² + b² + c² – 2ab – 2bc – 2ca  = 1/2[(a – b)²+(b – c)²+(c – a)²]
• a³ + b³ + c³ – 3abc =
  • = (a + b + c )(a² + b² + c² – ab – bc – ca )
  • = 1/2(a + b + c )[ (a – b)²+ (b – c)²+ (c – a)²]
  • = (a + b + c ) [ (a + b + c )² – 3( ab + bc + ca )]
  • Case -1 : if (a + b + c ) = 0,then a³ + b³ + c³ = 3abc 

Mensuration  2D

Triangle (त्रिभुज )

• Equilateral ( समबाहु त्रिभुज )

  • 

• Isoceles (समद्विबाहु त्रिभुज )

  • 

• Scalene (बिसमबाहु त्रिभुज )

  • Perimeter = a+ b+ c
  • Semi Perimeter (S) = (a+ b+ c)/2
  • Area = √[S (S-a ) (S-b )(S-c )]
• Right angle triangle (समकोण त्रिभुज )

• Right Isoceles triangle (त्रिभुज )

  • 

Square (वर्ग )

• Perimeter  =4a
• area = a² 
• Diagonal = = √2a

Rhombus (सम चतुर्भुज )

• Perimeter  =4a
• area = (d1 * d2)/2

Rectangle (आयत )

• Perimeter  =2 (a+ b)
• area = ab 
• Diagonal = = √(a² +b²⁾

Parallelogram (समान्तर चतुर्भुज )

• Perimeter  =2 (a+ b)
• area = ab 

Trapezium (सम्लब )

Circle 

Quadrilateral (चतुर्भुज)

Cyclic Quadrilateral (चक्रीय चतुर्भुज)

Bisentric Quadrilateral (द्विबीजपत्री चतुर्भुज)

Regular Octagon

Mensuration 3D

• Lateral/curved surface area (पार्श्व/वक्र पृष्ठीय क्षेत्रफल)
  • LSA/CSA = Perimeter of base * height
• Total surface area (कुल पृष्ठीय क्षेत्रफल)
  • TSA= LSA + 2(Area of base)
• Volume (आयतन ) 
  • V = Area of base * height 

Cube (घन)

• Side = a
• LSA/CSA = 4a²
• TSA = 6a²
• VOLUME = a³
• Side Diagonal = √2a
• Cube Diagonal (Longest Length) = √3a

Cuboid (घनाभ)

• Side = l, b, h
• CSA /LSA = Perimeter of Base * Height
  • 2 (l + b) * h
• TSA  = LSA +  2 (Area of Base)
  • 2(lb + bh + hl)
• Volume = Area of Base * Height
  • l*b*h
• Cuboid Diagonal (Longest Length) = √(l² + b² + h²)
• if X, Y, Z are three adjacent Surface Area of Cuboid Then
  • X =  lb , Y =  bh, Z = hl
  • Volume = √(XYZ)
• (l+b+h)² = l² + b² + h² +2(lb+bh+hl)
• (Sum of Side)² = (Diagonal)² + TSA

Cylinder (बेलन )

Cone (शंकु)

Sphere (गोला )

Hemi Sphere(अर्द्धगोला )

Tetrahedron (सम चतुष्फलक )

Prism (प्रिज्म )

Pyramid (पिरामिड )

Frustum ()

Kite (पतंग)