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Chapter 10 – Mensuration

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

Introduction

Everyday we come across different objects of various shapes. We see structured buildings, tools, etc. that are precisely sized and shaped according to different needs. But, have you ever wondered how are these objects made and why do they fit so perfectly in their use case?
People across the globe use certain measurement techniques to analyse the exact requirements and plan out on how to make a particular object.

Mensuration is that area of geometry that deals with the calculations related to different shapes and objects. The objects can be of different kinds, shapes, and purposes. In most of the cases, according to the characteristic of the object we want to analyse, we focus our calculations around length, area, and volume. To understand these concepts further, let us have a look at the theory of dimensions.

Dimensions can be seen as the total number of physical parameters that a particular object holds in space. For example- for a box, we have 3 parameters: its height, length, and breadth.

0-Dimension: An object is said to be 0-Dimensional if it has 0 physical parameters to measure. To understand it theoretically, think of a very small dot. We can’t define any of its physical parameters.

1-Dimension: An object is said to be 1-Dimensional if it has 1 physical parameter to measure. Now, let us take a 0-D dot that we defined earlier. Place ten of such dots in a line attached to each other. We will see a straight line, for which we can define the easiest physical parameter as its length.

2-Dimension: An object is said to be 2-Dimensional if it has 2 physical parameters to measure. Let us take a 1-D line that we defined earlier. Place ten of such lines side by side attached to each other. We will see a plane square area covered by those lines, for which we can define two physical parameters- length (made from dots) and breadth(made from lines).

3-Dimension: An object is said to be 3-Dimensional if it has 3 physical parameters to measure. Let us take a 2-D plane that we defined earlier. Place ten of such planes side on top of each other. We will see a box type volume made by those planes, for which we can define 3 physical parameters- length (made from dots), breadth(made from lines) and height(made from planes).

Let us say that we have defined the dot as 1 unit. Now taking this assumption, we will try to understand different measurements one by one.

Length 

Let us take a line. We know that line was made up of 10 dots joined together. Hence, according to the above assumption, we can say that the length of the line(=total number of dots in the line) is 10 units. This is because we are taking one dot as 1 unit, hence 10 dots together = 10 units.

Theoretically, length can be the physical parameter for a 1-D figure that gives us its spread in the space.

Area

Let us take a plane. We know that plane was made up of 10 lines joined together. Hence according to the above assumption, we can say that the total area covered by the plane(=total number of dots in the plane) is 10 x (number of dots in a line) units = 10 x 10 = 100.

Try to understand the areas of following figures- 

Theoretically, the area can be the physical parameter for 2-D figures that gives us its spread in the space.

Let us take a box now. We know that box(cube) was made up of 10 planes joined together. Hence according to the above assumption, we can say that the total volume covered by the cube(=total number of dots in the plane) is 10 x (number of dots in a plane) units = 10 x 10 = 100. 

Theoretically, the volume can be the physical parameter for 3-D figures that gives us its spread in the space.

Let us take an example of a square-shaped field, for which we need to apply 1 layer of the fence on the boundary. To do this, we need the total amount of fence that will be needed.

Total fence needed = length of face1 + length of face2 + length of face3 + length of face4. 

This length of the fence required is the perimeter of the field. Theoretically, the perimeter can be defined as the length of the boundary of any closed figure.

In this article, we discussed some of the fundamental concepts of mensuration like the theory of dimensions and major measurements that are required.

  1. What is a mensuration formula?
    A mensuration formula can be defined as a standard formula that can be used to compute the value of any physical parameter of a figure. Different formulas are used for different purposes and they entirely depend on various factors like the number of dimensions, the shape of the figure, etc.
  2.  What are the types of mensuration?
    Mensuration analysis and technique-s can be categorized on the basis of the number of parameters that are being analysed. Since the number of parameters in an analysis primarily depends on the number of dimensions in that particular figure, it basically boils down to dimensional categorization. Hence, there are majorly 3 types of mensuration- 1-D, 2-D and 3-D.
  3. How is mensuration used in our daily life?
    Mensuration is used in different analyses like distances, weights for different purposes, density, estimation of raw materials(example- fence in the above case), etc.
  4.  Who is the founder of mensuration?
    Archimedes is the founder of mensuration.

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