In this article, we will explain the important concepts and formulas with examples on 2D mensuration, which is an important topic in SSC CGL exam. 

What is mensuration?

It is defined as the branch of mathematics which deals with geometric shapes and figures, for calculation of their area, volume, perimeter and other related parameters. The literal meaning of mensuration is to measure. Let us define some important terms that will be used in this topic.

  • Area: It is a quantity that quantifies the extent of a 2D surface or shape in a plane. To understand area, think of a problem which asks you to find the amount of paint required to cover the surface with a single coat and the paint has a cost of some rupees or dollars per unit area. The units of area are  or .
  • Perimeter: It is the length of the path that surrounds an area. This term is used to quantify the length of the path. Speciically, the perimeter of a circle is called its circumference and is measured in m, cm or mm.
  • Volume: It is defined as the amount of space covered by a shape or a 3D object. The units of volume are  or .

Some basic unit conversions:

Units of area:

  • 1 Hectare = 10,000 m2
  • 1 km2 = 1000000 m2
  • 1 Decametre = 100 m2
  • 1 Decimeter square = 1/100 m2
  • 1 cm2 = 1/10000 m2
  • 1 mm2 = 1/1000000 m2

Units of volume:

  • 1 litre = 1000 cm3
  • 1 Hectametre3 = 10000 

 

Triangle:

 

Perimeter (P) = a +b + c

Area (A) = 

where  and a, b and c are three sides of the triangle. 

Also, , where b is base and h is the altitude.

Equilateral triangle:

Perimeter = 3a

Area (A) = , where 'a' is a side

Right angled triangle:

 

Area (A) =  and  (Pythagoras triplet), where p is perpendicular, b is base and h is hypotenuse.

Example: The base of a traingular field is 990 m and its height is 650 m. Calculate the area of the field. Also calculate the charges for supplying water to the field at the rate of INR 20.25 per sq.hectometre. 

Solution: Area of the field =     

Cost of supplying water to 1 sq. hm = INR 20.25

Cost of supplying water to the whole field = 20.25 x 32.175 = INR 651.54

Rectangle:

Example: The length and breadth of a rectangle are in the ratio 10:7. If its area is 700 sq. m., find its perimeter. 

Solution: Let the length and breadth of a rectangle are 10x and 7x respectively, in metres. For a rectangle, area = length x breadth

So, 700 = (10x)(7x), or    .

This gives, length = 10x = 10 x 3.162 = 31.62 m and breadth = 7x = 7 x 3.162 = 22.134 m.

Therefore, perimeter = 2(31.62+22.134)=107.5 m

Square:

Parallelogram:

Rhombus:

where a and b are the sides of triangle.

Irregular quadilateral:

 

Trapezium:

where a and b are length of two parallel sides, m and n are lengths of two non-parallel sides; h is the length of perpendicular to b.

Circle:

Semi circle:

Sector of a circle:

Area of sector = 

Length of an arc = 

Area of segment = Area of sector - Area of triangle = 

Perimeter of segment = length of the arc + length of segment =