Engineering Mathematics: Mensuration

By Mona Kumari|Updated : July 7th, 2021

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Important Formulas on Quadrilateral and Circle


A four-sided shape that is made up of two pairs of parallel lines and that has four right angles; especially: a shape in which one pair of lines is longer than the other pair.


The diagonals of a rectangle bisect each other and are equal.

Area of rectangle = length x breadth = l x b

OR Area of rectangle = byjusexamprep if one sides (l) and diagonal (d) are given.

OR Area of rectangle = image003 if perimeter (P) and diagonal (d) are given.

Perimeter (P) of rectangle = 2 (length + breadth) = 2 (l + b).

OR Perimeter of rectangle = byjusexamprep if one side (l) and diagonal (d) are given.


A four-sided shape that is made up of four straight sides that are the same length and that has four right angles.


The diagonals of a square are equal and bisect each other at 900.

 Area (a) of a square


Perimeter (P) of a square

= 4a, i.e. 4 x side


Length (d) of the diagonal of a square



A circle is a path travelled by a point which moves in such a way that its distance from a fixed point remains constant.


The fixed point is known as the centre and the fixed distance is called the radius.

(a) Circumference or perimeter of circle = image010

where r is radius and d is the diameter of the circle

(b) Area of circle

image011 is radius

image013  is circumference

image014  circumference x radius

(c) The radius of circle = image015


Sector :

A sector is a figure enclosed by two radii and an arc lying between them.



here AOB is a sector 

length of arc 


Area of Sector


Ring or Circular Path:

R=outer radius

r=inner radius





Rhombus is a quadrilateral whose all sides are equal.


The diagonals of a rhombus bisect each other at 900

Area (a) of a rhombus

= a × h, i.e. base × height

image019Product of its diagonals


since d2image021

since d2image022

Perimeter (P) of a rhombus

= 4a,  i.e. 4 x side


Where d1 and d2 are two-diagonals.

Side (a) of a rhombus



A quadrilateral in which opposite sides are equal and parallel is called a parallelogram. The diagonals of a parallelogram bisect each other.

Area (a) of a parallelogram = base × altitude corresponding to the base = b × h

Area of a parallelogram

Area (a) of the parallelogram byjusexamprep

where a and b are adjacent sides, d is the length of the diagonal connecting the ends of the two sides and image027


In a parallelogram, the sum of the squares of the diagonals = 2

(the sum of the squares of the two adjacent sides).

i.e., image029

Perimeter (P) of a parallelogram

= 2  (a+b),

Where a and b are adjacent sides of the parallelogram.

Trapezium (Trapezoid)

A trapezoid is a 2-dimensional geometric figure with four sides, at least one set of which are parallel. The parallel sides are called the bases, while the other sides are called the legs. The term ‘trapezium,’ from which we got our word trapezoid has been in use in the English language since the 1500s and is from the Latin meaning ‘little table.’


Area (a) of a trapezium

1/2 x (sum of parallel sides) x perpendicular 

Distance between the parallel sides

i.e., image031


Where,  l = b – a if b > a = a – b if a > b

And   image033

Height (h) of the trapezium


Pathways Running across the middle of a rectangle:


X is  the width of the path

Area of path= (l+b-x)x

perimeter=  2(l+b-2x)

Outer Pathways:




Inner Pathways:



Some useful Short trick:

  • If there is a change of X% in defining dimensions of the 2-d figure then its perimeter will also change by X%
  • If all the sides of a quadrilateral are changed by  X% then its diagonal will also change by X%.
  • The area of the largest triangle that can be inscribed in a semi-circle of radius r is r2.
  • The number of revolution made by a circular wheel of radius r in travelling distance d is given by

                          number of revolution =d/2πr

  • If the length and breadth of the rectangle are increased by x% and y% then the area of the rectangle will be increased by.


  • If the length and breadth of a rectangle are decreased by x% and y% respectively then the area of the rectangle will  decrease by:


  • If the length of a rectangle is increased by x%, then its breadth will have to be decreased by (100x/100+x)% in order to maintain the same area of the rectangle.
  • If each of the defining dimensions or sides of any 2-D figure is changed by x% its area changes by


where x=positive if increase and negative if decreases.


Important Mensuration (3D) Formulas



  • s = side
  • Volume: V = s^3
  • Lateral surface area = 4a2
  • Surface Area: S = 6s^2
  • Diagonal (d) = s√3



  • Volume of cuboid: length x breadth x width
  • Total surface area = 2 ( lb + bh + hl)

Right  Circular  Cylinder


  • Volume of Cylinder = π r^2 h
  • Lateral Surface Area (LSA or CSA) = 2π r h
  • Total Surface Area = TSA = 2 π r (r + h)



r1 = outer radius

r2 = inner radius


* Volume of Hollow Cylinder = π(pie) h(r1(Square) - r2(Square))

Right Circular Cone


  • l^2 = r^2 + h^2
  • Volume of cone = 1/3 π r^2 h
  • Curved surface area: CSA=  π r l
  • Total surface area = TSA = πr(r + l )

Important relation between radius, height and slant height of similar cone.



Frustum of a Cone

frustram cone

  • r = top radius, R = base radius,
  • h = height, s = slant height
  • Volume: V = π/ 3 (r^2 + rR + R^2)h
  • Surface Area: S = πs(R + r) + πr^2 + πR^2



  • r = radius
  • Volume: V = 4/3 πr^3
  • Surface Area: S = 4π^2



  • Volume-Hemisphere = 2/3 π r^3
  • Curved surface area(CSA) = 2 π r^2
  • Total surface area = TSA = 3 π r^2


Let 'r' is the radius of given diagram. You have to imagine this diagram, this is 1/4th part of Sphere.




  • Volume = Base area x height

prism 1

  • Lateral Surface area = perimeter of the base x height

prism 2



  • Volume of a right pyramid = (1/3) × area of the base × height.
  • Area of the lateral faces of a right pyramid = (1/2) × perimeter of the base x slant height.
  • Area of whole surface of a right pyramid = area of the lateral faces + area of the base.

1.From a solid cylinder no. of maximum solid cone of same height and radius as cylinder are 3.
2.From a solid sphere, no. of maximum solid cone having height and radius equal can be made are 4.
3.From a solid hemisphere, no. of maximum solid cone having height and radius equal can be made are 2.

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