Things to Remember: Important Formula for Geometry

Area of some rectilinear figures

  • Area of a Triangle : 1/2 x Base x Heights
  • Area of Equilateral Triangle : Sqrt(3)/4 x (Side)2
  • Area of a Triangle - Hero's formula :
    • Sqrt{s (s - a) (s - b) (s - c)} where 2s = a + b + c.
  • Area of Rectangle = Length x Breadth
  • Area of a Square = (Side)2
  • Area of Four Walls = 2 x height (Length + breadth)
  • Area of Rhombus = 1/2 x (Products of Diagonals)
  • Area of Trapezium = 1/2 x (Sum of Parallel Sides) x (Distance between Parallel lines)
  • Area of Parallelogram = Base x Altitude
  • Somestandards units of Area
    • 100mm2 = 1 cm2, 100 cm2, = 1 dm2
    • 100dm2 = 1 m2, 100 m2, = 1 are
    • 100arcs = 1 hactare
    • 100 hactare = 1 km2

Area related to Circles:

Circumference of a Circle or Perimeter of a Circle: The Distance arround the circle or the length of a circle is called its circumference or perimeter.

Circumference of a Circle or Perimeter of a Circle

 

Circumference (Perimeter) of a circle = d or 2r, where, r is the radius of the circle and = 22/7

Circle: If r be the radius of a circle, and its center is O.

Area if a circle, semi-circle and quadrant

 

Area of a Circle = r2

  Area of a Semi-Circle = r2
  Area of a Quadrant = r2

Perimeter of a Semi Circle: If r be the radius of a circle, and its center is O.

area and perimeter of a semi circle

  Perimeter of a Semi Circle : r + 2r

Area of a Ring: If r be the radius of a circle, and its center is O.

Area and perimeter of a ring

  Area of a ring or Annulus : (R2 - r2)

Length of arc:

length of an arc

  Length of arc AB = or

Area and Perimeter of a Sector :

area of a sector, sector area

  Area of a Sector OACBO = area of a sectoror (l x r)
  Area of a Sector OACBO = Length of arc AB + 2r

Area of a Segment :

area of a segment

  Area of a Minor Segemnt ACBA : Area of a sector OACBO - Area of Triangle OAB
= area of a sector - r2sin(theta)
  Area of a Major Segment BDAB : Area of a circle - Area of a minor segment ACBA
=r2 - Area of minor segment ACBA

Surface Area and Volumes:

Cuboid : If the length of a cuboid is l, breadth is b (sometimes we also use breadth as width 'w') and the height is 'h'.

surface area and volume of a cuboid

The formula for the lateral surface area, total surface area and its volume of a cuboid are

 

Lateral Surface area of a Cuboid= Area of four walls = 2 (length + breadth) x height =
= 2 (l + b) x h

 

Total Surface area of a Cuboid = 2 (length x breadth + breadth x height + height x length)
= 2 (lb + bh + hl)

 

Volume of a Cuboid = length x breadth x height = lbh

Cube : Basically a cube is a special type of a cuboid in which length, breadth and its height are equal i.e. length = breadth = height and it is also called an edge of a cube.

surface area and Volume of a cube

The formula for the lateral surface area, total surface area and its volume of a cube are

 

Lateral Surface area of a Cube = Area of four walls = 4 (edge)2

 

Total Surface area of a Cube = 6 x (edge)2

 

Volume of a Cuboid = (edge)3

Right Circular Cylinder : It the radius of a right circular cylinder is 'r' and its height is 'h'.

curved surface area, lateral surface area, Volume of a right cylinder cylinder

Then the curved surface area, total surface area and volume formula is given by

 

Area of the base of a cylinder = r2

 

Area of curved surface = Lateral surface area = Perimeter of the base x Height of cylinder

 

Total surface Area = 2rh + 2r2 = 2r (h + r)

Right Circular Hollow Cylinder : If the radius of internal and external cylinder is 'r' and 'R' respectively.

curved surface area and lateral surface area and Volume of a right circular hollow cylinder

Then the formula for curved and total surface area, volume of a hollow cylinder is given by

 

Area of curved surface = (Area of External Surface) + (Area of Internal Surface)
= 2Rh + 2rh

 

Total Surface Area = Curved surface area + 2 (Area of the base ring)
= (2Rh + 2rh) + 2(R2 - r2)
= 2(R + r) (h + R - r)

 

Volume of a hollow cylinder : Exterior Volume - Internal Volume
= (R2 h - r2h)
= h (R2 - r2)

Right Circular Cone: If r be the radius, h be the height and l be the slant height of a right circular cone.

curved surface area, lateral surface area, Volume of a right circular cone

Then the slant height, area of its curved surface and volume is given by the formula

 

Slant height (l) = √(h2 + r2)

 

Area of a Curved Surface = rl = r √(h2 + r2)

 

Total Surface area = Area of a curved surface + Area of a Base
= rl + r2
= r ( l + r)

 

Volume of a cone = 1/3 r2h

Sphere: If r be the radius of a sphere.

curved surface area, lateral surface area, Volume of the sphere

Then the formula for its surface area and its volume is given by

 

Surface Area = 4r2

 

Volume = 4/3 r2

Spherical shell : If r be the radius of internal, and R be the radius of external spherical shell.

curved surface area, lateral surface area, Volume of spherical shell

Then the surface area and volume is given by the formula

 

Curved Surface Area = 4R2

 

Volume = 4/3 R3 - 4 /3 r3= 4/3 (R3 - r3)

Hemisphere: If r be the radius of internal, and R be the radius of external spherical shell.

curved surface area and volume of a hemisphere

Then the surface area and volume is given by the formula

 

Curved Surface Area = 2r2

 

Total Curved Surface Area = Area of curved Surface + Area of its Base = 2r2 + r2 = 3r2

 

Volume = 2/3 r3

Frustum of a Cone : Fructum is a latin word and the meaning is 'piece cut off'. If any right circular cone is cut off by a plane, which is parallel to its base, then the portion of the cone between the plane and the base of a cone is known as a fractum of a cone.

area abd volume of a frustum

curved surface area, lateral surface area, Volume of frustum

Then the formula for curved and total surface area, volume of a frustum of a cone is given by

 

Curved Surface Area = l (R + r) where l = √[h2 + (R - r) 2]

 

Total Curved Surface Area = [R2 + r2 + l (R + r)]

 

Volume = 1/3 h [R2 + r2 + Rr]

  Here :
h = Vertical height of the frustum
l = slant height of the frustum
R = radius of the top
r = radius of the bottom
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