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SSC CGL Mensuration (2D) Complete Formulas and Basics

SSC CGL Mensuration (2D) Complete Formulas and Basics



It is very important to have an understanding of Different Formulas of quadrilaterals and circle for you to comfortably attempt Advanced Maths questions which covers a major portion of Quant Section of Competitive Exams. Here we are providing you formulas and shortcuts on how to solve mensuration questions.

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 =image002  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 (+ b).
OR Perimeter of rectangle =image004 if one sides (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.
(a) Area (a) of a square
image006
Perimeter (P) of a square
= 4a, i.e. 4 x side
image007
Length (d) of the diagonal of a square
image008
Circle
A circle is the path traveled 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 center and the fixed distance is called the radius.
(a) Circumference or perimeter of circle = image010
where r is radius and d is diameter of circle
(b) Area of circle
image011 is radius
image013  is circumference
image014  circumference x radius
(c) 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 AB= 2πrΘ/360°
Area of Sector ACBO=1/2[arc AB×radius]=πr×r×Θ/360°
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.
image017
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 parallelogram image026
where a and b are adjacent sides, d is the length of the diagonal connecting the ends of the two sides and image027
Untitled
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 changes by X%
  • If all the sides of a quadrilateral is changed by  X% then its diagonal will also changes 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 rectangle are increased by x% and y% then the area of the rectangle will increased by.
                                (x+y+xy/100)%
  • If the length and breadth of a rectangle is decreased by 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.

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