Which two figures out of the following four have the same area (with same units) ?

1. Fundamentals of 2D Mensuration (basic)

To master 2D mensuration, we must first understand the relationship between the boundaries of a shape and the space it occupies. A square is defined by its side length (s), with an area of s². A circle is defined by its radius (r), with an area of πr². In UPSC aptitude problems, you will often encounter 'composite figures'—shapes tucked inside others. A classic example is the 'packing' of circles within a square. For instance, in geography, we represent the Earth's orbit as circles of varying radii to visualize distances Science-Class VII . NCERT(Revised ed 2025), Earth, Moon, and the Sun, p.186, but in geometry, we look at how these shapes interact mathematically.

A fascinating principle arises when we pack identical circles into a square so that they touch the sides (tangent). Whether you place one large circle, four medium circles, or nine small circles, as long as they are identical and perfectly fill the square's width, the total area covered by the circles remains identical. This is because as the number of circles increases, their individual radii decrease proportionally. For a square of side s, the total area of the circles is always (πs²)/4. Consequently, the 'shaded' or 'leftover' area in the corners of the square remains constant regardless of how many circles you divide the space into.

Configuration	Radius of Each Circle	Total Area Formula	Simplified Total Area
1 Large Circle	s/2	1 × π(s/2)²	πs²/4
4 Identical Circles	s/4	4 × π(s/4)²	πs²/4
9 Identical Circles	s/6	9 × π(s/6)²	πs²/4

Understanding this proportionality is key to solving complex visualization problems. Just as great circles on a globe have the same length because they share the same circumference Certificate Physical and Human Geography, GC Leong, The Earth's Crust, p.14, these geometric partitions share the same area because their dimensions scale in a way that balances the total sum.

Sources: Science-Class VII . NCERT(Revised ed 2025), Earth, Moon, and the Sun, p.186; Certificate Physical and Human Geography, The Earth's Crust, p.14

2. Relationship between Dimensions and Area (basic)

Welcome to the second step of our journey! To master quantitative aptitude, we must first understand that Area is a two-dimensional measure of a surface. While linear dimensions like length or radius are one-dimensional, area depends on the square of these dimensions. A fascinating principle in geometry is that the total area of a set of shapes can remain constant even if we change their number, provided the dimensions are adjusted proportionally.

Let's consider a practical example: packing circles inside a square of side s. If you place one large circle that touches all four sides, its diameter is s and its radius is s/2. Using the formula Area = πr², the area of this circle is π(s/2)² = πs²/4. Now, what if we replace that one circle with four smaller, identical circles? Each small circle would have a diameter of s/2 and a radius of s/4. The area of one small circle is π(s/4)² = πs²/16. Since there are four such circles, their total area is 4 × (πs²/16), which simplifies back to πs²/4. This proves that the total area covered by the circles—and consequently the remaining "empty" space in the square—remains identical regardless of whether you have 1, 4, 9, or 16 circles, as long as they are packed symmetrically.

This concept of proportional dimensions is vital across disciplines. In physics, for instance, we see a fixed relationship in spherical mirrors where the radius of curvature R is exactly twice the focal length f (R = 2f) Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.137. Similarly, in geography, we understand that while the Earth is a sphere, the density of a population is a ratio of the number of people to the specific unit of land area they occupy Fundamentals of Human Geography, Class XII (NCERT 2025 ed.), The World Population Distribution, Density and Growth, p.8. Whether we are measuring the 3.28 million square km of India Contemporary India-I, Geography, Class IX, India Size and Location, p.1 or the area of a Great Circle on the globe Certificate Physical and Human Geography, GC Leong, The Earth's Crust, p.14, the underlying geometry of how dimensions define space remains the same.

Sources: Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.137; Fundamentals of Human Geography, Class XII (NCERT 2025 ed.), The World Population Distribution, Density and Growth, p.8; Contemporary India-I, Geography, Class IX, India Size and Location, p.1; Certificate Physical and Human Geography, GC Leong, The Earth's Crust, p.14

3. Understanding Composite Figures (intermediate)

In quantitative aptitude, Composite Figures are shapes formed by combining two or more basic geometric forms like squares, circles, or triangles. To master these, we must look beyond the visual complexity and identify the underlying proportionality. A classic principle involves inscribed circles within a square. If you have a square of side s, the area of a single circle tangent to all four sides is π(s/2)². Interestingly, if you replace that one circle with four smaller, identical circles arranged in a 2x2 grid, the total area occupied by the circles remains exactly the same. This is because while the radius of each small circle is halved (s/4), the number of circles is quadrupled (4), perfectly balancing the calculation: 4 × π(s/4)² = πs²/4.

This concept of Area Conservation is vital when analyzing spatial data. Just as we calculate the total geographical area of India as 3.28 million sq km NCERT, Contemporary India II, p.7, we often need to partition that area into specific segments, such as the 57% forest cover found in the Peninsular Plateau and Hills Environment and Ecology, Major Crops and Cropping Patterns in India, p.80. In geometry problems, the 'shaded region' (the empty space between the circles and the square's corners) serves as a metaphor for these partitions; no matter how many identical circles you pack into the square (1, 4, 9, or 16), the total shaded area remains a constant fraction of the square's total area.

Configuration	Total Circle Area Calculation	Result
1 Large Circle	π × (s/2)²	(πs²)/4
4 Small Circles	4 × [π × (s/4)²]	(πs²)/4
9 Smaller Circles	9 × [π × (s/6)²]	(πs²)/4

Sources: NCERT, Contemporary India II, Resources and Development, p.7; Environment and Ecology (Majid Hussain), Major Crops and Cropping Patterns in India, p.80

4. Scaling and Proportionality in Geometry (intermediate)

In geometry, proportionality is a powerful tool that allows us to understand how changing one dimension affects another. When we deal with 2D shapes, we must remember that area is a function of the square of the linear dimensions. For instance, just as we measure the volume of a cuboid by multiplying its length, width, and height as discussed in Science, Class VIII, p.145, we calculate the area of a circle using its radius squared (πr²). A fascinating application of this principle occurs when we 'pack' shapes into a fixed container, such as a square of side 's'.

Imagine a square containing one large circle that touches all four sides. The radius is s/2, and the area is π(s/2)². Now, if we replace that one circle with four smaller, identical circles arranged in a 2x2 grid, each circle now has a radius of s/4. While each individual small circle has only 1/4th the area of the large one, there are four of them. Mathematically, 4 × π(s/4)² simplifies exactly to πs²/4. This demonstrates that the total area occupied by the circles remains constant regardless of whether you have 1, 4, 9, or 16 circles, provided they are packed proportionally to fill the square. This is similar to how the elasticity of a demand curve is determined by the ratio of its segments, as seen in Microeconomics, Class XII, p.30; in geometry, it is the constant ratio between the circle and its circumscribing square (which is always π/4) that dictates the result.

This principle of scaling is essential for competitive exams because it allows you to solve complex-looking 'shaded region' problems through logic rather than tedious calculation. Whether you are looking at a map of Singapore or New York to estimate distances from the equator Geography, GC Leong, p.10, or analyzing geometric partitions, the underlying rules of scale and proportion ensure that if the ratio of the parts to the whole remains the same, the total area remains unchanged. This is why different configurations of circles within the same square will always leave the exact same amount of 'empty' or shaded space in the corners.

Sources: Science, Class VIII (NCERT), The Amazing World of Solutes, Solvents, and Solutions, p.145; Microeconomics, Class XII (NCERT), Theory of Consumer Behaviour, p.30; Certificate Physical and Human Geography, GC Leong, The Earth's Crust, p.10

5. Symmetry and Pattern Recognition (intermediate)

In quantitative aptitude, Symmetry and Pattern Recognition are powerful tools that allow us to solve complex-looking problems by identifying underlying geometric invariants. At its core, symmetry is about balance and repetition. When a figure is divided into identical smaller parts, the properties of the whole can often be determined by analyzing a single representative unit. For instance, just as a rectangular pattern in a settlement consists of straight lanes meeting at right angles to create a predictable grid Geography of India, Majid Husain, Settlements, p.6, geometric patterns in aptitude often follow strict mathematical proportions that keep total values constant even as the complexity of the visual layout increases.

A classic application of this principle is the Area Conservation in packed circles. Imagine a square with side length s. If you place one large circle inside it that touches all four sides, its area is π(s/2)². Now, if you instead pack that same square with four smaller identical circles (arranged in a 2x2 grid), the radius of each small circle becomes s/4. You might think the area changes, but if you calculate the total area—4 × π(s/4)²—it simplifies exactly to πs²/4. This shows that the shaded region (the empty space between the circles and the square) remains identical regardless of whether you have 1, 4, 9, or even 100 circles packed symmetrically within.

This concept of proportional scaling is vital. In the same way that the nature and size of an image in a mirror depend on the object's position relative to the center of symmetry Science, Class X, Light – Reflection and Refraction, p.137, geometric areas depend on the ratio of dimensions. When we scale down a shape (like a circle) but increase its frequency (the number of circles) by the square of that scale factor, the total area remains constant. Recognizing these patterns saves a student from tedious calculations during an exam.

Configuration	Number of Circles	Radius of Each	Total Area
Single Large	1 (1²)	s / 2	πs² / 4
Medium Grid	4 (2²)	s / 4	πs² / 4
Small Grid	9 (3²)	s / 6	πs² / 4

Sources: Geography of India, Majid Husain, Settlements, p.6; Science, Class X (NCERT), Light – Reflection and Refraction, p.137

6. Circle Packing and Area Invariance (exam-level)

In quantitative aptitude, the concept of Circle Packing and Area Invariance is a fascinating intersection of geometry and scaling. Imagine a square with a side length of s. If we place a single large circle inside it that touches all four sides, its diameter is s and its radius is s/2. The area of this circle is π(s/2)² or (πs²)/4. Now, if we replace that one circle with four smaller, identical circles arranged in a 2×2 grid, each circle now has a diameter of s/2 and a radius of s/4. While the circles are smaller, there are more of them. The total area becomes 4 × π(s/4)², which simplifies exactly to (πs²)/4. Remarkably, the total area occupied by the circles remains constant regardless of whether you pack 1, 4, 9, or 16 identical circles inside.

This principle teaches us that the total area of a set of figures is often determined by the proportionality of their dimensions to the bounding container. Just as we might compare the sizes of continents by counting squares to understand their relative areas Exploring Society: India and Beyond, Oceans and Continents, p.36, we can see here that the "density" of the circles within the square is invariant. In every case, the circles will always occupy approximately 78.5% (π/4) of the square's area, leaving the remaining "shaded" area in the corners and gaps to be exactly the same total size.

Understanding this invariance is crucial because it simplifies complex-looking problems. If a question asks you to compare the "wasted space" or the shaded region between a square and 16 small circles versus a square and 1 large circle, you don't need to perform exhaustive calculations. Because the underlying dimensions are proportional, the area of the gaps remains identical. This is a geometric constant that holds true as long as the circles are identical and packed perfectly to the edges of the square.

Sources: Exploring Society: India and Beyond (NCERT Class VI), Oceans and Continents, p.36; Certificate Physical and Human Geography (GC Leong), World Population, p.295

7. Solving the Original PYQ (exam-level)

You've just mastered the concepts of Area Calculation for Composite Figures and Geometric Symmetry, and this PYQ is the perfect test of how those "building blocks" come together. The core logic relies on the principle that the total area of circles inscribed within a square remains constant, regardless of how many identical circles you pack in, provided they are tangent to the sides and each other. As you learned in the module on Mensuration and Plane Figures, the total area of n circles with radius r/n within the same boundary will always simplify back to the original area of a single inscribed circle.

Let’s walk through the reasoning as a coach would: In Figure 1, a single circle in a square of side s has an area of π(s/2)². In Figure 3, where the square is divided into four smaller quadrants, each contains a circle with a radius of s/4. When you calculate the total area—4 × π(s/4)²—it simplifies exactly to (πs²)/4. Because the area occupied by the circles is mathematically identical, the remaining space (the shaded region) within the square must also be identical. This confirms that 1 and 3 share the same area, making (A) 1 and 3 the correct answer.

UPSC often sets "visual traps" by including options like 2 or 4 to exploit the illusion of density. A student might intuitively feel that having four circles covers more ground than one large circle, or that a different configuration changes the total surface area. However, as we saw in our study of Geometric Proportions, the increase in the number of circles is perfectly balanced by the squared decrease in their individual radii. Don't let the complexity of the "packing" distract you from the Scaling Laws; if the proportions are consistent, the total area remains invariant.