Discover How Mathematics Helps Calculate Plot Areas Accurately

The importance of mathematics cannot be emphasised enough in our ever-evolving world. From calculating the trajectory of space missions to designing sustainable architectural structures, mathematics is the language that empowers us to understand and navigate the complexities of our surroundings. Surprisingly, this fascinating subject extends its reach even to the seemingly traditional realm of land measurement and property assessment.

The concepts that are relevant to this article are typically covered in mathematics curricula of Class 9 to Class 12, depending on the educational board of the school. These concepts are generally found in chapters related to geometry and trigonometry. In this article, we will explore how mathematics plays a crucial role in enabling Patwaris to accurately calculate plot areas.

Who Is A Patwari?

A Village Accountant, also known as Patwari in Madhya Pradesh, Punjab, Telangana, West Bengal, Karanam in Andhra Pradesh, Talati in Gujarat, Karnataka, Maharashtra, and Lekhpal in Uttar Pradesh, Uttarakhand, is a government position found in rural areas. The role of Patwaris in land measurement is of paramount importance, especially in rural areas where land ownership is a significant aspect of people's lives. Patwaris are responsible for maintaining land records, ensuring the accuracy of measurements, and updating them when necessary. They act as custodians of crucial data that forms the foundation of land ownership and property transactions.

To accurately determine the area of a plot, Patwaris employ a range of mathematical techniques, formulas, and tools. These calculations are not mere exercises in arithmetic; rather, they are the means by which land boundaries are marked and property values assessed. By understanding these mathematical principles, students can gain a deeper insight into the practical applications of the subject and appreciate its relevance in everyday life.

Importance Of Accurate Plot Measurements

Imagine you're planning to buy or sell a plot of land. In such a scenario, determining the exact area of the plot becomes a critical factor. Even a small error in the measurement can lead to significant discrepancies in land transactions, resulting in financial losses and legal disputes. To ensure fair and transparent land transactions, patwaris, who are land record officers, come into play. These professionals specialise in land measurements and play a vital role in accurately determining plot areas.

Role Of Mathematics In Land Measurements

But how do patwaris calculate plot areas with such precision? The answer lies in the realm of mathematics. Patwaris rely on various mathematical concepts and techniques to carry out their measurements effectively. By exploring these mathematical foundations, we can unravel the intricate process of plot area calculations.

Geometry: Key to Accurate Measurements

Geometry, the branch of mathematics that deals with shapes, sizes, and their properties, plays a crucial role in land measurements. Patwaris use geometry extensively to measure the dimensions of plots accurately. They apply geometric formulas to calculate the area of irregularly shaped plots, such as rectangles, triangles, and circles. By understanding the fundamentals of geometry, patwaris can decipher complex land boundaries and calculate their areas with precision.

Trigonometry: Unlocking Inaccessible Areas

Trigonometry, the study of relationships between angles and sides of triangles, is another indispensable tool in the patwaris' arsenal. In certain cases, plots may have inaccessible areas or angles that cannot be measured directly. However, by utilising trigonometric principles, patwaris can overcome these challenges and obtain accurate measurements.

Patwaris employ trigonometric functions like sine, cosine, and tangent to determine distances or heights within plots. By measuring known distances and angles, they can calculate the length of inaccessible sides or the height of elevated portions. These measurements enable them to accurately determine the plot's area, even in complex situations where direct measurements are not feasible.

Surveying Techniques: Enhancing Accuracy

Patwaris also rely on surveying techniques that heavily rely on mathematical principles. They use instruments such as theodolites and total stations, which employ mathematical calculations to measure angles and distances. These tools enable patwaris to obtain precise readings, which are crucial for calculating plot areas.

The theodolite, for instance, measures horizontal and vertical angles between points on the plot. By applying trigonometry, the readings obtained from the theodolite can be used to determine the lengths of sides and angles within the plot. Similarly, the total station combines Electronic Distance Measurement (EDM) with angular measurements to provide highly accurate data for calculating plot areas. The proper use of these surveying techniques, coupled with mathematical calculations, ensures reliable and precise measurements.

Coordinate Systems: Locating Plots with Precision

Patwaris often work with land records that are plotted on maps or land survey sheets. To determine the area of a plot accurately, they employ coordinate systems such as latitude and longitude or Cartesian coordinates. By understanding coordinate systems and their mathematical underpinnings, patwaris can precisely locate plots on maps and accurately calculate their areas.

Using coordinate systems, patwaris can identify the coordinates of various boundary points of the plot. They then connect these points to form a closed figure, employing geometric and trigonometric principles to calculate the area enclosed by the boundary. By leveraging these mathematical tools, patwaris can ensure precise measurements even for plots located in vast areas or complex geographical landscapes.

Area Calculation Of Regular Shapes

The ability to calculate the area of regular shapes is highly beneficial for a Patwari, as it enables them to accurately determine the size and dimensions of plots of land. Following are some regular shapes.

Area Of Parallelogram

To find the area of a parallelogram, we can use a clever trick. We can imagine cutting the parallelogram and rearranging its parts to create a rectangle. The neat thing is that the parallelogram and the rectangle will have the same area!

In the picture, you can see that the parallelogram and the rectangle have the same base length (the blue line) and the same height (the green line). The base length is the length of one of the sides of the parallelogram, and the height is how tall the parallelogram is.

We know that the area of a rectangle is found by multiplying its base length and height. So, we can use the same measurements to find the area of the parallelogram: base length multiplied by height.

No matter where we cut the parallelogram, as long as it's perpendicular to the base, the base length will remain the same. It's like we can slide the cut along the base and still get the same area.

So, to calculate the area of a parallelogram, all we need to do is multiply its base length by its height. That's it! It's a simple and easy way to find the area of a parallelogram using the same formula as a rectangle.

Area = length width

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Area Of Triangle

We can break down a triangle into two parts by cutting it parallel to the base. This slicing creates a triangle and a trapezoid. Interestingly, if we make the cut right in the middle of the height, these two parts fit together to form a parallelogram with the same base but half the height.

In the image, you can see how this dissection works. The base remains the same, but the height is halved. By multiplying the base by half of the height, we can find the area of the triangle. Similarly, if we multiply half of the base by the full height, we also get the same result. These calculations simplify to the formula

Area = 1/2× base × height

Which represents the area of the triangle.

Area = √{s(s-a)(s-b)(s-c)}

Where s = (a+b+c)/3

And a, b, c are sides of a triangle.

Area Of Trapezoid

Similar to the triangle, we can also combine two copies of a trapezoid to form a parallelogram.

In the image, you can see how the two trapezoids fit together to create a parallelogram. The height of the parallelogram is the same as the height of the trapezoid, while its base is the sum of the two bases of the trapezoid.

Therefore,

Area of the parallelogram = height × (base₁ + base₂).

However, it's important to note that the area we obtained represents two trapezoids combined. To find the area of just one trapezoid, we need to cut the total area in half. This is because the parallelogram encompasses two identical trapezoids, so dividing the area by two gives us the area of a single trapezoid.

So, to calculate the area of a trapezoid, we use the formula:

1/2× height × (base₁ + base₂)

This formula takes into account the height and the sum of the two bases of the trapezoid, allowing us to determine its area accurately.

Area Of circle

Area of the circle is calculated by πr² where r is the radius of the circle.

Solved Problem

Consider the following plot with an irregular shape. Calculate the total area of the plot.

Solution:

The total area of the figure is given as:

⇒ Area = Area (ABIM) + Area (BCGH) + Area (CDEF) + Area (JKL)

⇒ Area = (AB × BI) + (BC × CG) + (CD × DE) + (12× LJ × KO)

⇒ Area = ( 10 × 5) + (3 × 3) + (2 × 2) + (1/2× 4 × 4)

⇒ Area = 50 + 9 + 4 + 8

⇒ Area = 71 cm²

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Hope you have now a good understanding of how mathematics empowers patwaris to accurately calculate plot areas.