Quantitative Aptitude : Mensuration & Geometry

By Asha Gupta|Updated : October 18th, 2021

Complete coverage of syllabus is a very important aspect for any competitive examination but before that important subject and their concept must be covered thoroughly. In this article, we will cover Quantitative Aptitude : Mensuration & Geometry

Mensuration & Geometry

Triangle

A triangle is the simplest polygon enclosed by three sides. As in the figure below three sides AB, BC, and CA represent triangle ABC or denoted by ∆ABC. A B and C represent the vertices of the ∆ABC. Also, a, b and c represent the lengths of the sides BC, AC and AB respectively and ÐA, ÐB and ÐC represent the values of angles of the ∆ABC.

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Classification of Triangles:

1.  On the basis of Sides of the Triangle

Scalene Triangle: If all the sides of a triangle are of unequal lengths then the triangle is termed as Scalene Triangle. In the figure, Δ ABC is a scalene triangle.

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Equilateral Triangle: If all the three sides of a triangle are equal in length then the Triangle is called an Equilateral Triangle. If the length of the side is ‘a’.

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 Isosceles Triangle: If any two sides of a triangle are of equal length “a” unit and the third side is of “b” unit length then the triangle is said to be an Isosceles Triangle.

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 2.     On the basis of Angle of a Triangle:

Obtuse-angled Triangle: In a triangle, if one angle is more than 90° or the other two angles are less than 90° then the triangle is said to be an obtuse-angled.

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Acute-angled Triangle: In a triangle, if all the angles are less than 90° then the triangle is said to be an acute-angled triangle.

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Right Angled Triangle: If in a triangle, two sides are perpendicular to each other or make 90° to each other then the triangle is said to be a Right-Angled Triangle. Here in the figure Δ ABC, AB is perpendicular, BC is base and AC is the hypotenuse of the Right-Angled triangle.

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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 side (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 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

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Length (d) of the diagonal of a square

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Circle

A circle is a path travelled by a point that 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 a 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 

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Area of Sector

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Ring or Circular Path:

R=outer radius

r=inner radius

image005 

area=π(R2-r2)

Perimeter=2π(R+r)

Rhombus

The rhombus is a quadrilateral whose all sides are equal.

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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

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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 tricks:

  • 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 revolutions made by a circular wheel of radius r in travelling distance d is given by

                          the 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.

 

Important Mensuration

Cube

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

Cuboid

  • 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)

Hollow-Cylinder

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r1 = outer radius

r2 = inner radius

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* 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.

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Frustum of a 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

Sphere

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

Hemisphere

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

Quarter-Sphere

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

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Prism

  • Volume = Base area x height
  • Lateral Surface area = perimeter of the base x height

Pyramid

  • The 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 the whole surface of a right pyramid = area of the lateral faces + area of the base.

 

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