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Chapter 11 – Mensuration

The following Topics and Sub-Topics are covered in this chapter and are available on MSVgo:

Introduction

Imagine that you want to purchase a carpet for your room, how would you know a perfect size? Simple, you would need to measure the area which you wish to cover. Similarly, if you want to fence your garden, you need to measure the entire length or distance around the boundary.

In both these examples, we are talking about the measurement of Perimeter, length, and area of different geometrical shapes. The field of mathematics dealing with all these parameters is called Mensuration.

Previously you have studied Perimeter and area of known and familiar shapes and before we continue, let us revise what we already know.

  • Geometric shapes are of two types –
    • 2D shapes like Rectangle, Square, Circle, Parallelogram, Triangle, Trapezium.
    • 3D shapes have volume and surface area like Cube, Cuboids, Cylinder, Pyramid.
  • The area is the entire region covered by the given shape, whereas Perimeter is the total length of a given shape. For example,
    Area of square = a2 and Perimeter = 4 x a

    Area of rectangle = a x b and Perimeter = 2(a+ b)

    Area of Triangle = ½ b x h and Perimeter = sum of all the three sides

    Area of circle = 𝜋𝑟2 and Perimeter = 2𝜋𝑟, where r is the radius

    Area of Parallelogram = b x h and Perimeter = sum of all the sides

A quadrilateral with two parallel sides is known as trapezium. If the non-parallel sides are equal, it becomes isosceles trapezium.

You can find the area of the trapezium by dividing the figure into two or three plane figures; for example, we have divided the trapezium into two triangles and one rectangle.

You know how to find the area of triangle and rectangle which makes it easy for you to find the area of trapezium.

A = Area of triangle 1 + Area of triangle 2 + Area of rectangle

 = ½ hx + ½ hy + ah

 = ½ h (x + y + 2a), we know that total breadth, b = (x + y + a)

Therefore, the area of trapezium = 1/2h (a + b) or 1/2h (sum of both parallel sides)

First, draw a diagonal in the quadrilateral to divide it into two triangles, as shown below. After that draw perpendiculars h1 and h2 to AC.

Now, the area of quadrilateral ABCD will be the sum of areas of both the triangles.

A = (area of ∆ ADC) + (area of ∆ ABC)

    = (1/2 h1 * AC) + (1/2 h2 * AC)

    = ½ AC (h1 + h2), AC = d where d denotes the length of diagonal AC.

Therefore, Area of the quadrilateral = ½ d (h1 + h2)

There is no exact formula to calculate the area of polygons. However, you can figure it by splitting the given shape in possible known figures and then adding those areas.

For example, you can calculate the area of the octagon by dividing it into two trapeziums and one rectangle and adding their areas.

Area of octagon = Area of trapezium 1 + Area of trapezium 2 + Area of rectangle

Three-dimensional objects occupying some space and having length, breadth and depth or height are called solid shapes.

Some shapes have two or more identical or congruent faces, for instance, cube.

You can find the surface area of any solid by adding the areas of its faces.

  • Surface Area of a Cube

    You know that cube has six identical faces. Therefore, the surface area of a cube = 6a2
  • Surface Area of a Cuboid

    A cuboid has three pairs of identical faces.
    Therefore, the surface area of a cuboid = 2(ab + bc + ca), where a, b, and c are the length, width and height, respectively.
  • Surface Area of a Cylinder
    You must have seen a cold drink can. It is cylindrical in shape and consists of three surfaces: two parallel circles (top and bottom) and side.

    When you unroll the sides of the can, you will see that it is a rectangle.

    Area of the Rectangle part = L x W = 2πrh.
    You can add this area to the areas of two circles to find the total surface area of the cylinder.Therefore A = 2πr2 + 2πrh, where r and h are the radius and height, respectively.
  • Volume Of Cube Cuboid And Cylinder
    The amount of region occupied by any three-dimensional object is known as its volume.Volume of Cube
    Volume is the product of length, height and breadth and in a cube, all three are equal. Hence, the Volume of Cube = a3.

    Volume of cuboid
    Similar to cube, the volume is the product of length, breadth and height.

    Hence, Volume of Cuboid = = l * b * h
  • Volume of Cylinder
    Cylinders and rectangular solids are quite similar as both have a height and two bases.
    You can use the rectangular solid to find the formula for the volume of the cylinder. For rectangle, V = Base * Height, here height = h and base = l * wHence, V = l * w * h
    Similarly, for Cylinder, V = Base * Height, here height = h and base = πr2 . Hence, Volume of Cylinder = πrh
  • Volume indicates the total space occupied by a 3D object and measured in cubic units like cubic meters. 
  • The capacity of an entity refers to the quantity it can hold and can be measured in gallons, pounds, litres and more. For example, if an oil tank stores 50000 litres oil, then its capacity is 50000 litres.
  • What is mensuration formula?
    There is no specific mensuration formula. Different shapes have a different formula for example the area of a square is a2, and that of a rectangle is axb.
  • What is called Mensuration?
    It is the study of the measurement of parameters like area, volumes, perimeters of various 3D and 2D geometric shapes.
  • What are the types of Mensuration?
    a. 2D Mensuration – that deals with 2-dimensional shapes like circle, square, rectangle, triangles and more.
    b. 3D Mensuration – that deals with 3-dimensional figures like a cylinder, cube, cuboid, sphere, and more.

  • Why do we study Mensuration?
    There are many real-life applications of Mensuration, for example, for packaging milk and other food items, measurement of volumes is crucial.
  • What is the use of Mensuration?
    There are many real-life applications of Mensuration.
  • Who is the founder of Mensuration?
    Archimedes is the founder of Mensuration.To understand and learn more about Mensuration, visit MSVgo – a specially curated video library app packed with explanatory and interactive visualizations. We believe in clearing concepts instead of following the rules and formulas blindly, so that learning becomes fun and entertaining.

Mensuration is a significant topic, as these concepts and formulas also have real-life application. For example, if you know the area of the walls, you can very easily determine how much paint is needed to paint the entire wall. 

In this chapter, you have explored relationships between different geometrical shapes and also learned to apply mathematical formulas. For instance, If you want to wrap a rectangular pencil box, then you can calculate how much paper is required by finding the surface area of that box.

High School Physics

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Middle School Math

  • Addition
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