A rectangular plot of lawn shown in the figure has dimensions x and y and is surrounded by a gravel pathway of width 2 m. What is the total area of the pathway ?

1. Foundations of 2D Geometry: Rectangles and Squares (basic)

In quantitative aptitude, mastering 2D geometry begins with understanding the rectangle and the square. A rectangle is a quadrilateral where opposite sides are equal and all internal angles are 90°. If all four sides are equal, it becomes a square. To visualize this, consider a school playground that is 40 m in length and 30 m in width Exploring Society: India and Beyond, Locating Places on the Earth, p.10. The space inside is the Area (Length × Width), and the distance around it is the Perimeter (2 × (Length + Width)).

A frequent challenge in competitive exams involves calculating the area of a pathway or border added around a shape. Imagine a rectangular lawn with length x and width y. If we build a uniform path of width 2 m outside the lawn, we must account for this addition on all sides. Because the path is added to both the left and right sides, the new total length becomes x + 2 + 2 = x + 4. Similarly, because the path is added to both the top and bottom, the new total width becomes y + 2 + 2 = y + 4.

To find the Area of the Path alone, we use a simple subtraction logic:

• Step 1: Calculate the Area of the larger Outer Rectangle: (x + 4)(y + 4).
• Step 2: Calculate the Area of the original Inner Lawn: (x × y).
• Step 3: Subtract the inner area from the outer area: (x + 4)(y + 4) − xy.

This principle of "Outer minus Inner" applies whether you are looking at a small garden or conceptualizing large-scale regional structures, such as the segments of the Golden Quadrilateral Geography of India, Transport, Communications and Trade, p.3. Expanding the algebraic expression (x + 4)(y + 4) − xy gives us 4x + 4y + 16, which represents the precise surface area of the path.

Sources: Exploring Society: India and Beyond. Social Science-Class VI . NCERT(Revised ed 2025), Locating Places on the Earth, p.10; Geography of India ,Majid Husain, (McGrawHill 9th ed.), Transport, Communications and Trade, p.3

2. Understanding Area as a Product of Dimensions (basic)

To understand area, we must first view it as the measure of a two-dimensional surface. While a line has only one dimension (length), a shape like a rectangle occupies space across two directions: length and width. The area is simply the product of these two dimensions. For instance, a school playground with a length of 40 m and a width of 30 m has an area of 1,200 m² Exploring Society: India and Beyond, Locating Places on the Earth, p.10. On a much larger scale, the landmass of India is measured in square units, totaling approximately 3.28 million square km Contemporary India-I, India Size and Location, p.1. Units of area are always 'squared' (m², km², or hectares) because they represent the product of two linear units.

A critical application of this concept in competitive exams involves changing dimensions. Imagine a rectangular lawn with length x and width y. If we decide to build a walking path of a uniform width (let's say 2 meters) around the outside of this lawn, the dimensions of the entire figure change. Because the path surrounds the lawn, it adds 2 meters to both ends of the length and both ends of the width. Therefore, the new length becomes x + 2 + 2 = x + 4, and the new width becomes y + 2 + 2 = y + 4. The area of this new, larger rectangle is the product of these expanded dimensions: (x + 4)(y + 4).

To find the area of just the path itself, we use the logic of subtraction of regions. We calculate the area of the 'outer' rectangle and subtract the area of the 'inner' rectangle (the original lawn). This leaves us only with the area of the border. This principle is widely used in geography and land planning to determine 'reporting areas' and land use classes India People and Economy, Geographical Perspective on Selected Issues and Problems, p.111.

Sources: Exploring Society: India and Beyond, Locating Places on the Earth, p.10; Contemporary India-I, India Size and Location, p.1; India People and Economy, Geographical Perspective on Selected Issues and Problems, p.111

3. Units and Measurement Consistency (basic)

At its heart, measurement consistency is the golden rule of quantitative aptitude: you must always compare 'apples to apples.' In any mathematical or scientific problem, units represent the physical reality of a number. If you attempt to add, subtract, or multiply numbers with different units—such as adding meters to kilometers without conversion—the resulting value loses all physical meaning. As noted in the study of density, the units we use depend entirely on the units of mass and volume we choose; for example, the SI unit of density is kg/m³ because it is derived from the SI units for mass (kg) and volume (m³) Science, Class VIII, p.141.

When solving geometry or geography problems, pay close attention to the scale of measurement. For instance, while a village field might be measured in local units like bighas, the standard international unit for land area is the hectare, which is defined as the area of a square with a side of 100 meters Economics, Class IX, p.3. If a problem provides the dimensions of a plot where one side is in meters and another is in kilometers, your first step must always be to convert them to a single uniform unit. This principle applies even to vast measurements, such as India's total area of 3.28 million square kilometers India Physical Environment, Geography Class XI, p.5.

In more complex fields like optics or meteorology, consistency is the safeguard against error. When calculating the magnification of an image, if the object height and focal length are given in centimeters (cm), every other distance in that calculation must also stay in centimeters to ensure the ratio remains valid Science, Class X, p.156. Similarly, atmospheric pressure can be expressed in millibars, kilograms per cm², or inches of mercury Certificate Physical and Human Geography, Weather, p.117. While these units are interchangeable through conversion, mixing them within a single equation without a conversion factor is a common trap in competitive exams.

Sources: Science, Class VIII, The Amazing World of Solutes, Solvents, and Solutions, p.141; Economics, Class IX, The Story of Village Palampur, p.3; India Physical Environment, Geography Class XI, India — Location, p.5; Science, Class X, Light – Reflection and Refraction, p.156; Certificate Physical and Human Geography, Weather, p.117

4. Algebraic Variables in Geometry (intermediate)

In competitive aptitude, we move from arithmetic (using fixed numbers) to algebraic geometry, where we use variables like x and y to represent dimensions. This approach is powerful because it allows us to derive a general formula for any scenario. For instance, when dealing with a rectangular region (like a lawn or a room) of length x and width y, the area is simply xy. However, complexity arises when we add a uniform border or pathway around that region. Logic dictates that if a path of width w is added to all sides, it extends the length by w on the left and w on the right, making the new length x + 2w. Similarly, the width becomes y + 2w. This algebraic substitution is a standard method for solving equilibrium and growth problems, much like how variables are used to determine market equilibrium in Microeconomics, Market Equilibrium, p.74. To find the area of just the pathway, we calculate the Difference of Areas: the 'Outer Rectangle' minus the 'Inner Lawn.' Algebraically, this is expressed as (x + 2w)(y + 2w) − xy. When we expand this product, we get xy + 2wx + 2wy + 4w². After subtracting the original lawn area (xy), we are left with the expression 2wx + 2wy + 4w². For a path that is 2 meters wide (where w = 2), the formula simplifies beautifully to 4x + 4y + 16. Mastering these expansions is essential because it transforms a visual geometry problem into a solvable linear equation, a technique frequently used to derive multipliers or sums in other quantitative fields, as seen in Macroeconomics, Money and Banking, p.51.

Sources: Microeconomics, Market Equilibrium, p.74; Macroeconomics, Money and Banking, p.51

5. Practical Applications: Fencing and Border Costs (intermediate)

When we discuss fencing and border costs, we are essentially looking at the geometry of boundaries. Whether we are installing "live fences" for livestock protection Environment and Ecology, Majid Hussain, p.26 or developing a "green belt" around an industrial area Environment, Shankar IAS Academy, p.81, the mathematical principle remains the same: The Frame Logic. To calculate the area of a border (like a path around a garden) or the length of a fence, we must distinguish between the inner dimensions and the outer dimensions.

Imagine a rectangular plot with length L and width W. If you construct a pathway of uniform width x outside this plot, the pathway adds to the plot on all sides. This means you add the width x to the left and the right (Length becomes L + 2x), and to the top and the bottom (Width becomes W + 2x). This is similar to drawing an outline around a rectangular object, such as a glass slab, to define its total boundary Science, Class X NCERT, p.147. The total area of the region including the path is found by multiplying these new dimensions, much like calculating total revenue in economics by finding the area of a rectangle (Price × Quantity) Microeconomics, NCERT Class XII, p.59.

Feature	Calculation Type	Key Formula Component
Fencing	Perimeter (Linear)	2 × (Length + Width)
Paving/Pathways	Area (Surface)	Outer Area − Inner Area
Costing	Rate Application	Total Units × Cost per Unit

To find the Area of the Border alone, you subtract the area of the original plot from the area of the total outer rectangle. If the plot is 10m × 8m and the path is 2m wide, the outer dimensions are 14m × 12m.
Path Area = (14 × 12) − (10 × 8) = 168 − 80 = 88 m². Understanding this "doubling" of the width (adding to both sides) is the most critical step in avoiding errors in competitive exams.

Sources: Environment and Ecology, Majid Hussain, Locational Factors of Economic Activities, p.26; Environment, Shankar IAS Academy, Environmental Pollution, p.81; Science, Class X NCERT, Light – Reflection and Refraction, p.147; Microeconomics, NCERT Class XII, The Theory of the Firm under Perfect Competition, p.59

6. Geometry of Paths: Internal and External Boundaries (exam-level)

In quantitative aptitude and field geometry, understanding the relationship between internal and external boundaries is fundamental. When we talk about a path surrounding a rectangular area—such as a garden, a playground, or a settlement—the geometry is defined by how the 'width' of that path interacts with the 'dimensions' of the inner space. As noted in Geography of India, Majid Husain, Settlements, p.6, rectangular patterns often feature lanes that meet at right angles, which allows us to use standard algebraic formulas to calculate area. To find the area of a path, we use the Subtraction Method: Area of Path = (Outer Rectangle Area) - (Inner Rectangle Area). If we have a rectangular lawn with length x and width y, and we add a path of width w around the outside, the new dimensions are not simply x + w and y + w. Because the path exists on both sides (left/right and top/bottom), we must add the width twice. The new length becomes x + 2w and the new width becomes y + 2w. This is a critical step in precise land measurement, a practice that has maintained the form of Indian rural settlements for centuries Geography of India, Majid Husain, Settlements, p.7. Let’s look at the algebra for an external path of 2 meters around an x by y lawn. The outer area is (x + 4)(y + 4). When we expand this, we get xy + 4x + 4y + 16. To isolate just the path, we subtract the original lawn area (xy). The remaining expression, 4x + 4y + 16, represents the area of the path alone. This logic is equally applicable when mapping school playgrounds or calculating the scale of a local map Exploring Society: India and Beyond, NCERT Class VI, Locating Places on the Earth, p.10.

Scenario	New Length	New Width	Logic
External Path (width w)	L + 2w	B + 2w	Path is added to both ends of the original dimension.
Internal Path (width w)	L - 2w	B - 2w	Path is carved out from the inside of the original dimension.

Sources: Geography of India, Settlements, p.6; Geography of India, Settlements, p.7; Exploring Society: India and Beyond, NCERT Class VI, Locating Places on the Earth, p.10

7. Solving the Original PYQ (exam-level)

Building on the principles of 2D Mensuration and Algebraic Expansion you have just mastered, this question tests your ability to visualize spatial relationships and apply the 'Difference of Areas' method. The building blocks come together here by treating the lawn and the path as two nested rectangles. In UPSC CSAT Quantitative Aptitude, these problems require you to translate a verbal description into a geometric model where the pathway is the region between the outer and inner boundaries.

To arrive at the correct answer, visualize the expansion of the dimensions: because the 2m path surrounds the lawn on all sides, you must add 2m to both the left and right (making the new length $x+4$) and 2m to both the top and bottom (making the new width $y+4$). The total area including the path is $(x+4)(y+4)$, which expands to $xy + 4x + 4y + 16$. By subtracting the original lawn area ($xy$), the $xy$ terms cancel out, leaving you with 4x + 4y + 16. This logical sequence confirms that Option (D) is the only mathematically sound result.

UPSC often includes 'distractor' options to catch students who rush their logic. Options (A) and (B) are common traps designed for those who might only add the path width to one side of the rectangle or confuse the area calculation with the perimeter. Option (C) is particularly clever; it accounts for the side strips but ignores the four corner squares. Each corner of the path is a $2 \times 2$ square ($4 m^2$), and since there are four corners, they add exactly 16 to the total area. Recognizing these 'corner overlaps' is the hallmark of a prepared candidate.