Important Mensuration (2D) Formulas, Mathematics Study Notes

By Neha Joshi|Updated : November 12th, 2020

In this article, we will be covering a very important topic from Geometry that is – Important Mensuration (2D) Formulas. This article will hold importance for all the candidates who are preparing for CTET, UPTET, TREET, MPTET or any other TET exam. The notes are prepared by experts who have deep insight into the examination syllabus and pattern, let's get started.

Important Formulas on Quadrilateral and Circle

Rectangle

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.

image001

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.

Square

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

image005

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

 Area (a) of a square

image006

Perimeter (P) of a square

= 4a, i.e. 4 x side

byjusexamprep

Length (d) of the diagonal of a square

byjusexamprep

Circle

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

image009

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

image016

Sector :

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

image003

 

here AOB is a sector 

length of arc 

byjusexamprep

Area of Sector

byjusexamprep

Ring or Circular Path:

R=outer radius

r=inner radius

image005 area=π(R2-r2)

Perimeter=2π(R+r)

Rhombus

Rhombus is a quadrilateral whose all sides are equal.

byjusexamprep

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

image020

since d2image021

since d2image022

Perimeter (P) of a rhombus

= 4a,  i.e. 4 x side

image023

Where d1 and d2 are two-diagonals.

Side (a) of a rhombus

image024

Parallelogram

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

byjusexamprep

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.’

image030

Area (a) of a trapezium

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

Distance between the parallel sides

i.e., image031

image032

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

And   image033

Height (h) of the trapezium

image034

Pathways Running across the middle of a rectangle:

image006

X is  the width of the path

Area of path= (l+b-x)x

perimeter=  2(l+b-2x)

Outer Pathways:

image007

Area=(l+b+2x)2x

Perimeter=4(l+b+2x)

Inner Pathways:

Area=(l+b-2x)2x

Perimeter=4(l+b-2x)

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.

                                (x+y+xy/100)%

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

                                    (x+y-xy/100)%

  • 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

          x(2+x/100)%

where x=positive if increase and negative if decreases.

Thanks

byjusexamprep

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