Consider the following figures : Which one of the following conclusion can be drawn from these figures ?

1. Introduction to 2D Mensuration and Plane Figures (basic)

Concept: Introduction to 2D Mensuration and Plane Figures

2. Properties of Quadrilaterals: Squares and Rectangles (basic)

To master quantitative aptitude, we must first understand the building blocks of geometry: quadrilaterals. A rectangle is a four-sided polygon where every interior angle is a right angle (90°). In practical terms, rectangles are the standard for spatial organization, seen in everything from school playgrounds to the ancient planned patterns of rural settlements in India Exploring Society: India and Beyond. Social Science-Class VI, Locating Places on the Earth, p.10. A Square is simply a special type of rectangle where all four sides are equal in length. Because they share this right-angle property, both shapes are fundamental to calculating Area, which is the measure of the surface inside the boundaries. In economic models, for instance, we represent total revenue and total cost as the area of rectangles formed by price/cost and quantity Microeconomics (NCERT class XII 2025 ed.), The Theory of the Firm under Perfect Competition, p.58.

The properties of these shapes allow us to derive consistent formulas for measurement. For a rectangle with length (l) and width (w), the Area = l × w. For a square with side (s), the Area = s². Beyond area, we often need to find the Diagonal, which is the straight line connecting opposite corners. Using the Pythagorean theorem, the diagonal of a rectangle is √(l² + w²). This geometric precision has historically influenced how land is measured in India; for example, the Bigha system often maintains rectangular or square field patterns that dictate the layout of entire villages Geography of India, Settlements, p.7.

Comparison of Properties:

Feature	Rectangle	Square
Opposite Sides	Equal and Parallel	Equal and Parallel
All Sides	Not necessarily equal	Always equal
Interior Angles	All 90°	All 90°
Diagonals	Equal; Bisect each other	Equal; Bisect each other at 90°

Sources: Exploring Society: India and Beyond. Social Science-Class VI, Locating Places on the Earth, p.10; Microeconomics (NCERT class XII 2025 ed.), The Theory of the Firm under Perfect Competition, p.58; Geography of India, Settlements, p.7

3. Basics of Triangles and Height-Base Relationship (basic)

At its core, a triangle is the simplest polygon, formed by three straight lines meeting at three points. In quantitative aptitude, the most critical relationship to master is how we calculate the space it occupies—its Area. To understand this, we must define two key terms: the Base and the Height (also known as the Altitude). While any of the three sides can serve as the base, the height must always be the perpendicular distance from that base to the opposite vertex. Just as the term 'altitude' describes the vertical height of a mountain or a plane above a reference level Social Science-Class VI: Exploring Society, Landforms and Life, p.42, the altitude of a triangle represents its vertical 'reach' relative to its base.

The fundamental formula for the area of a triangle is:
Area = ½ × Base × Height

Why the '½'? Imagine a rectangle with a specific width and length. If you draw a diagonal line through it, you split it into two identical triangles. Therefore, a triangle with a base of b and a height of h will always occupy exactly half the area of a rectangle with those same dimensions. This relationship is a constant in geometry; as long as the product of the base and height remains the same, the area remains unchanged, regardless of the triangle's specific shape or orientation.

Sources: Social Science-Class VI: Exploring Society, Landforms and Life, p.42

4. Concept of Perimeter: The Boundary Length (intermediate)

At its core, Perimeter is the total length of the continuous line forming the boundary of a closed geometric figure. Imagine you are standing at one corner of a school playground and walk along the fence until you return to your starting point; the total distance you travelled is the perimeter. This concept is vital in practical fields, from calculating the amount of fencing needed for a garden to determining the length of a wire required for an electrical circuit. While area measures the surface inside a shape, perimeter focuses strictly on the one-dimensional boundary. For instance, when we map a rectangular playground of 40 m length and 30 m width, the perimeter represents the sum of all four outer edges Exploring Society: India and Beyond, Locating Places on the Earth, p.10. To calculate the perimeter of any polygon, you simply sum the lengths of all its sides. However, for regular shapes, we use specific formulas to speed up our calculations. For a square, since all four sides (s) are equal, the perimeter is 4s. For a rectangle, which has two equal lengths (l) and two equal widths (w), the formula is 2(l + w). In the context of physics, the length of a wire (essentially its perimeter if laid out) directly affects its physical properties, such as resistance Science, Electricity, p.180. Understanding that different shapes can share the same perimeter while having different internal areas is a hallmark of mastering intermediate geometry.

Feature	Perimeter	Area
Definition	Total length of the boundary.	Total space enclosed within the boundary.
Dimension	One-dimensional (Linear).	Two-dimensional (Square units).
Unit	m, cm, km.	m², cm², km².

Sources: Exploring Society: India and Beyond, Locating Places on the Earth, p.10; Science, Electricity, p.180

5. Units of Measurement and Dimensional Analysis (intermediate)

To master quantitative aptitude, we must first understand that every number in a problem represents a physical reality defined by Units and Dimensions. At the most fundamental level, we have Base Units like mass (kilogram), length (metre), and time (second). However, most quantities we deal with in geometry and physics are Derived Units—they are combinations of these base units. For example, Area is a derived unit calculated as Length × Length, giving it the dimension of L² (Length squared). Whether you are calculating the area of a square (side²), a rectangle (length × breadth), or a triangle (½ × base × height), the resulting unit will always be square units (e.g., m²), allowing us to compare different shapes on a single numerical scale. Science, Class VIII, Pressure, Winds, Storms, and Cyclones, p.82

Moving beyond simple geometry, Dimensional Analysis helps us understand complex relationships. Consider Pressure, defined as Force per unit Area. Its SI unit is N/m², also known as the Pascal (Pa). Science, Class VIII, Pressure, Winds, Storms, and Cyclones, p.87 Similarly, Density depends on the units of mass and volume chosen; while the SI unit is kg/m³, we often use g/mL or g/cm³ for convenience in laboratory settings. Science, Class VIII, The Amazing World of Solutes, Solvents, and Solutions, p.141 The core rule is Dimensional Homogeneity: you can only add, subtract, or directly compare quantities if they have the same dimensions. You cannot subtract 5 kilograms from 10 metres because their dimensions (Mass vs. Length) do not match.

Sometimes, we use Conversion Factors to translate different physical units into a common "equivalent" unit for comparison. A prime example is in environmental science, where various greenhouse gases are converted into CO₂ equivalent using their Global Warming Potential (GWP). Environment, Shankar IAS Academy, Environment Issues and Health Effects, p.425 This allows us to represent the impact of methane or nitrous oxide in a single, standardized unit, much like how we use a standard unit of area to compare a square to a triangle. In your aptitude tests, always check if the units are consistent before performing calculations—mixing cm and m, or grams and kilograms, is the most common path to an incorrect answer.

Sources: Science, Class VIII, NCERT, Pressure, Winds, Storms, and Cyclones, p.82, 87; Science, Class VIII, NCERT, The Amazing World of Solutes, Solvents, and Solutions, p.141; Environment, Shankar IAS Academy, Environment Issues and Health Effects, p.425

6. Isoperimetric Concepts: Comparing Area and Perimeter (intermediate)

In geometry, the relationship between Area (the space inside) and Perimeter (the boundary length) is not fixed; shapes with the exact same area can have very different perimeters. This is a fundamental concept in spatial reasoning. Just as we measure the dimensions of a physical object like a notebook or a box to find its volume Science, Class VIII NCERT, The Amazing World of Solutes, Solvents, and Solutions, p.145, we must systematically calculate the specific metrics of 2D shapes to compare them accurately rather than relying on visual intuition. Consider three distinct shapes that all occupy the same amount of 'space' (Area = 36 square units):

• Square: With a side of 6 units, the area is 6 × 6 = 36. Its perimeter is 4 × 6 = 24 units.
• Rectangle: With sides of 9 and 4 units, the area is 9 × 4 = 36. However, its perimeter is 2(9 + 4) = 26 units.
• Triangle: With a base of 12 and a height of 6, the area is ½ × 12 × 6 = 36. Depending on its specific side lengths, its perimeter will generally be even larger than the square's.

This comparison leads us to the Isoperimetric Principle. It teaches us that for a fixed area, a circle is the most 'efficient' shape (having the smallest perimeter), followed by a square. As a rectangle becomes longer and thinner, its perimeter increases even if the area remains constant. In the UPSC CSAT or competitive aptitude exams, recognizing that equal area does not imply equal perimeter is crucial for solving complex geometry and mensuration problems quickly.

Sources: Science, Class VIII NCERT, The Amazing World of Solutes, Solvents, and Solutions, p.145

7. Area Calculation Formulas for 2D Shapes (exam-level)

In the UPSC CSAT and other quantitative exams, understanding Area is fundamental. Area represents the measure of the surface enclosed by a 2D boundary, expressed in square units (like cm² or m²). While shapes may look entirely different to the eye, their "footprint" or area can be identical if the product of their dimensions remains the same.

Let’s break down the three most common shapes you will encounter:

• The Square: A square is a regular quadrilateral where all sides are equal. To find its area, we simply square the length of its side (Side × Side). For instance, if you are cutting a square hole in a cardboard box with a side of 6 cm, the area of that opening is 36 cm² Science-Class VII . NCERT, Light: Shadows and Reflections, p.163.
• The Rectangle: A rectangle is defined by its length (l) and width (w). The area is the product of these two dimensions (Length × Width). This logic is used everywhere, from calculating the size of a playground Exploring Society: India and Beyond. Social Science-Class VI . NCERT, Locating Places on the Earth, p.10 to visualizing economic concepts like Total Revenue, which is represented as the area of a rectangle formed by Price and Quantity Microeconomics (NCERT class XII), The Theory of the Firm under Perfect Competition, p.58.
• The Triangle: The area of a triangle is exactly half that of a rectangle with the same base and height. The formula is ½ × Base × Height. It is vital to remember that the height must be the perpendicular distance from the base to the opposite vertex.

Shape	Dimension Requirements	Area Formula
Square	Side (s)	s²
Rectangle	Length (l), Width (w)	l × w
Triangle	Base (b), Vertical Height (h)	½ × b × h

A critical takeaway for the exam is that different dimensions can yield the same area. A square with side 6, a rectangle of 9 × 4, and a triangle with base 12 and height 6 all result in an area of 36 units. When solving comparison questions, always calculate the numerical value rather than relying on visual intuition.

Sources: Science-Class VII . NCERT, Light: Shadows and Reflections, p.163; Exploring Society: India and Beyond. Social Science-Class VI . NCERT, Locating Places on the Earth, p.10; Microeconomics (NCERT class XII), The Theory of the Firm under Perfect Competition, p.58

8. Analyzing Geometric Figures through Calculation (exam-level)

In quantitative aptitude, Analyzing Geometric Figures requires moving beyond visual intuition to objective measurement. We often encounter diverse shapes—squares, rectangles, and triangles—that look entirely different but may share identical spatial properties. Just as geographic settlements can be identified by their geometric patterns, such as the Rectangular Pattern where lanes meet at right angles Geography of India, Settlements, p.6, we must identify the specific mathematical properties that define a shape's area.

To compare figures effectively, we rely on three fundamental formulas. When we calculate these, we are essentially performing a comparative analysis, similar to how economists compare different poverty estimation metrics like URP and MRP to find a common ground for evaluation Indian Economy, Poverty, Inequality and Unemployment, p.37. The core formulas you must master are:

• Square: Area = side × side (or s²). Since all sides are equal, a single measurement defines the entire space.
• Rectangle: Area = length × breadth. This applies to any four-sided figure where opposite sides are equal and meet at 90-degree angles.
• Triangle: Area = ½ × base × height. Note that the 'height' must be the perpendicular distance from the base to the opposite vertex.

It is crucial to remember that the physical appearance or orientation of a shape does not change its area. Much like how the shadow of an opaque object might change shape or size depending on the light's angle Science-Class VII, Light: Shadows and Reflections, p.158, the numerical area of a geometric figure remains constant regardless of how it is rotated or positioned on a page. In exam scenarios, always calculate the area for each figure individually before drawing a conclusion about their relationship.

Sources: Geography of India, Settlements, p.6; Indian Economy, Poverty, Inequality and Unemployment, p.37; Science-Class VII, Light: Shadows and Reflections, p.158

9. Solving the Original PYQ (exam-level)

Now that you have mastered the fundamental formulas for Area and Perimeter, this CSAT question serves as the perfect application of those building blocks. In the UPSC, questions often look deceptively simple or visually complex to test your conceptual clarity rather than just your calculation speed. Here, you must synthesize your knowledge of three distinct shapes—a square, a rectangle, and a triangle—and objectively compare their properties rather than relying on how they appear to the eye.

Let’s walk through the calculations as a disciplined aspirant would. For Figure I (Square), with a side of 6, the area is 6 × 6 = 36. Moving to Figure II (Rectangle), the area is length × breadth (9 × 4), which also equals 36. Finally, for Figure III (Triangle), applying the formula ½ × base × height (½ × 12 × 6), we again arrive at 36. Because each calculation yields the same numerical value, we can confidently conclude that (B) The areas of all the three figures are equal. This exercise proves that shapes with entirely different dimensions and visual profiles can occupy the exact same amount of space.

It is equally important to understand why the other options are classic UPSC traps. Options (C) and (D) focus on perimeter, which is a common distractor designed to catch students who confuse the two concepts or fail to perform the second set of calculations. The perimeter of the square is 24 units, while the rectangle's perimeter is 26 units (2 × [9+4]); thus, they are not equal. Option (A) is a "visual trap" intended for candidates who make assumptions based on the height or width of the figures without applying the formulas. Remember: always verify with data rather than intuition when dealing with geometric comparisons.