A rectangular piece of iron sheet measuring 50 cm by 100 cm is rolled into a cylinder of height 50 cm. If the cost of painting the cylinder is Rs. 50 per square metre, then what will be the cost of painting the outer surface of the cylinder ?

1. Basics of 2D Mensuration: Areas of Quadrilaterals (basic)

To master 2D mensuration, we must first understand the Quadrilateral — a closed flat shape with four straight sides and four vertices. The term 'Area' refers to the amount of surface enclosed within these boundaries. In a geographical context, for instance, the total land mass of India is described as having an area of 3.28 million square km CONTEMPORARY INDIA-I, India Size and Location, p.1. Understanding how to calculate these areas allows us to quantify everything from a small school playground to vast national territories. The most fundamental quadrilateral is the Rectangle. Its area is calculated by multiplying its two dimensions: Area = Length × Width. For example, if a playground is 40 m long and 30 m wide, its area is 1,200 square meters Exploring Society: India and Beyond, Locating Places on the Earth, p.10. A Square is a special type of rectangle where all four sides are equal; thus, its area is simply Side². When measuring these areas, consistency in units is vital—if lengths are in meters (m), the area must be in square meters (m²). Moving beyond right-angled shapes, we encounter the Parallelogram and the Trapezium. A parallelogram's area is determined by its Base × Perpendicular Height (Area = b × h). A trapezium, which has only one pair of parallel sides, requires a slightly different approach: we take the average of the two parallel sides and multiply it by the height between them: Area = ½ × (Sum of Parallel Sides) × Height. These basic formulas are the building blocks for surveying land use and understanding geographical data Contemporary India II, The Rise of Nationalism in Europe, p.7.

Quadrilateral	Key Property	Area Formula
Rectangle	Opposite sides equal; 90° angles	Length × Width
Square	All sides equal; 90° angles	Side × Side (s²)
Parallelogram	Opposite sides parallel	Base × Height
Trapezium	One pair of parallel sides	½ × (a + b) × h

Sources: CONTEMPORARY INDIA-I, India Size and Location, p.1; Exploring Society: India and Beyond, Locating Places on the Earth, p.10; Contemporary India II, The Rise of Nationalism in Europe, p.7

2. Circle Geometry and the Concept of Circumference (basic)

At its most fundamental level, a circle is a collection of all points in a plane that are at a fixed distance from a central point. This fixed distance is known as the radius (r). When you draw a line from one edge of the circle to the other, passing through the center, you get the diameter (D). As a rule of thumb, the diameter is always twice the length of the radius (D = 2r). This geometric principle is so universal that it even applies to the physics of light; for instance, in spherical mirrors, the radius of curvature is twice the focal length Science Class X NCERT, Light – Reflection and Refraction, p.137.

The circumference is essentially the perimeter of the circle—the total distance around its outer boundary. Imagine taking a piece of string, wrapping it perfectly around a circular object, and then straightening it out to measure it with a ruler. That linear length is the circumference. In geography, we see this concept applied to the Earth; the equator is the largest possible circle (a Great Circle) that can be drawn around the globe, representing its maximum circumference Certificate Physical and Human Geography, The Earth's Crust, p.14. When visualizing the Earth's orbit or distances, we often use specific radii to represent these circular or near-circular paths Science-Class VII NCERT, Earth, Moon, and the Sun, p.186.

The magic of a circle lies in the constant relationship between its circumference and its diameter. No matter how large or small the circle is, if you divide its circumference by its diameter, you always get the same number: π (pi), which is approximately 3.14159 or 22/7. This gives us the two standard formulas for circumference:

• Circumference = π × Diameter
• Circumference = 2 × π × radius

Understanding this formula is vital because it allows us to "unroll" a curved shape into a straight line, which is a key skill in solving quantitative aptitude problems involving cylinders or rolling objects.

Sources: Science Class X NCERT, Light – Reflection and Refraction, p.137; Certificate Physical and Human Geography, The Earth's Crust, p.14; Science-Class VII NCERT, Earth, Moon, and the Sun, p.186

3. Surface Areas of 3D Solids: The Cylinder (intermediate)

To master the geometry of a cylinder, we must first visualize how it is constructed. Imagine a standard rectangular sheet of paper. When you roll this sheet such that two opposite edges meet, you create a 3D shape with a curved surface. In a laboratory setting, you often see this shape in a measuring cylinder, where the liquid inside also forms a unique curved surface called a meniscus Science, Class VIII, NCERT, The Amazing World of Solutes, Solvents, and Solutions, p.144. In mathematical terms, a right circular cylinder consists of two congruent parallel circular bases joined by a curved surface.

The Curved Surface Area (CSA)—sometimes called the lateral surface area—is simply the area of that "unrolled" rectangle. When you unroll a cylinder of height (h) and radius (r):

• The height of the rectangle is the same as the height of the cylinder (h).
• The length of the rectangle is equal to the distance around the circular base, which is its circumference (2πr).

Therefore, the formula is derived from basic area logic: Area = Length × Breadth, which gives us CSA = 2πrh.

When we talk about the Total Surface Area (TSA), we are accounting for every square inch of the solid's exterior. This includes the curved side plus the two flat circular ends. Since the area of one circle is πr², for two bases we add 2πr². This gives us the complete picture of the solid's boundary.

Component	Formula	Visual Logic
Curved Surface Area	2πrh	Circumference of base × Height
Area of Two Bases	2πr²	Two flat circular "lids"
Total Surface Area	2πr(r + h)	Sum of CSA and both bases

In competitive exams, pay close attention to units. Surface area is always measured in square units (e.g., cm², m²). A common pitfall is forgetting that 1 m² is equal to 10,000 cm² (since 100 cm × 100 cm = 10,000 cm²). Always convert your dimensions to a uniform unit before applying these formulas.

Sources: Science, Class VIII, NCERT, The Amazing World of Solutes, Solvents, and Solutions, p.144

4. Metric System and Unit Conversion Scaling (intermediate)

The Metric System is a decimal-based system of measurement used globally, built on powers of ten. In competitive exams, the challenge isn't just knowing that 1 meter equals 100 centimeters; it is understanding how these units scale when we move from linear dimensions (length) to two-dimensional space (area) or three-dimensional space (volume). As noted in Understanding Economic Development, DEVELOPMENT, p.12, even simple health metrics like the Body Mass Index (BMI) require precise unit conversion—converting height from centimeters to meters before squaring it—to ensure the final value is accurate.

The most common mistake students make is applying a linear conversion factor to an area or volume problem. For example, if you are calculating the area of a land mass like India, which is approximately 3.28 million square km (CONTEMPORARY INDIA-I, India Size and Location, p.1), you are dealing with squared units. The fundamental rule of scaling is: if the linear conversion factor is k, the area conversion factor is k², and the volume conversion factor is k³.

Dimension	Relationship	Example (m to cm)	Conversion Factor
Linear (Length)	1 m = 100 cm	10 m = 1,000 cm	10¹
Area (2D)	1 m² = (100 cm)²	1 m² = 10,000 cm²	10²
Volume (3D)	1 m³ = (100 cm)³	1 m³ = 1,000,000 cm³	10³

When solving aptitude problems involving costs (e.g., "Rs. 50 per square meter"), you must ensure your area units match the rate units. If your calculation results in 5,000 cm², you cannot simply divide by 100. You must divide by 10,000 (which is 100 × 100) to get 0.5 m². Similarly, when calculating volume for scientific purposes, such as multiplying length, width, and height (Science Class VIII, The Amazing World of Solutes, Solvents, and Solutions, p.145), the resulting units will be in cubic terms (cm³), requiring a division by 1,000,000 to reach cubic meters (m³).

Sources: Understanding Economic Development, DEVELOPMENT, p.12; CONTEMPORARY INDIA-I, India Size and Location, p.1; Science Class VIII, The Amazing World of Solutes, Solvents, and Solutions, p.145

5. Arithmetic Applications: Rates, Costs, and Proportions (basic)

At the heart of many quantitative aptitude problems lies the concept of a rate—a ratio that compares two different quantities. In economic terms, this is often expressed as an Average Cost (AC), which is simply the total cost divided by the number of units produced or area covered. For instance, if you know the total cost of painting a wall, you can find the rate by dividing that cost by the total area. As noted in Microeconomics (NCERT class XII 2025 ed.), Production and Costs, p.44, the short-run average cost (SAC) is defined as the total cost per unit of output, a principle that applies whether you are manufacturing widgets or calculating the price of tiling a floor.

When dealing with costs, it is vital to distinguish between fixed and variable components. A fixed cost remains constant regardless of the scale (like the price of a brush), while a variable cost changes in direct proportion to the quantity (like the amount of paint used). In competitive markets, we often look for the break-even point, where the revenue generated exactly covers these costs Microeconomics (NCERT class XII 2025 ed.), Market Equilibrium, p.88. Understanding these proportions allows us to scale a small observation—like the cost of one square centimeter—up to a large project.

A common pitfall in arithmetic applications is unit conversion. Rates are frequently provided in one unit (e.g., Rupees per square meter), while measurements are given in another (e.g., centimeters). Because area is a two-dimensional measure, scaling is not linear. For example, since 1 meter equals 100 centimeters, 1 square meter (1m × 1m) equals 10,000 square centimeters (100cm × 100cm). Failing to account for this squared relationship is a frequent source of error in competitive exams.

Concept	Definition	Mathematical Formula
Rate (Average Cost)	Cost incurred per single unit of measurement.	Total Cost / Total Units
Proportional Scaling	The increase in output resulting from a proportional increase in all inputs.	Output ∝ Input
Area Conversion	Converting linear units to square units for cost application.	(Linear Factor)²

Sources: Microeconomics (NCERT class XII 2025 ed.), Production and Costs, p.44; Microeconomics (NCERT class XII 2025 ed.), Market Equilibrium, p.88

6. Geometric Transformations: Rolling a Sheet into a Cylinder (exam-level)

To understand how a flat sheet transforms into a 3D cylinder, we must look at the geometry of a rolled surface. Imagine taking a rectangular piece of chart paper—as suggested for building a cylindrical coil in Science, Class VIII, NCERT (Revised ed 2025), Electricity: Magnetic and Heating Effects, p.49. When you roll this sheet, the 2D surface area of the paper doesn't disappear; it simply wraps around to become the Curved Surface Area (CSA) of the cylinder.

The transformation follows two fundamental rules:

• One side of the rectangle becomes the height (h) of the cylinder.
• The other side of the rectangle wraps around to become the circumference (C) of the cylinder's base.

Because the area of a rectangle is Length × Breadth, the area of the cylinder's curved wall is Circumference × Height (or 2πrh). If you are given a sheet that, when rolled, has a circumference of 100 cm and a height of 50 cm, its curved surface area is simply 100 cm × 50 cm = 5000 cm².

A critical step in aptitude exams is unit consistency. Often, dimensions are given in centimeters (cm), but costs or final answers are required in meters (m). To convert area from cm² to m², we must remember that 1 m = 100 cm. Therefore, 1 m² = 100 cm × 100 cm = 10,000 cm². To convert 5000 cm² to m², we divide by 10,000, resulting in 0.5 m². This precision is vital, just as reading the meniscus accurately is essential when measuring volumes in a cylinder Science, Class VIII, NCERT (Revised ed 2025), The Amazing World of Solutes, Solvents, and Solutions, p.144.

Sources: Science, Class VIII, NCERT (Revised ed 2025), Electricity: Magnetic and Heating Effects, p.49; Science, Class VIII, NCERT (Revised ed 2025), The Amazing World of Solutes, Solvents, and Solutions, p.144

7. Solving the Original PYQ (exam-level)

This problem is a classic application of Dimensional Transformation, where a 2D rectangle is mapped onto a 3D cylinder. You have just mastered how the length of a sheet becomes the circumference of the cylinder while the breadth remains the height (or vice versa). The core conceptual link here is that the Curved Surface Area (CSA) of the cylinder is numerically identical to the area of the original rectangular sheet, as no material is added or removed during the rolling process. Visualizing the rectangle 'wrapping' around an axis is the bridge between flat geometry and solid mensuration.

To arrive at the correct answer, we first identify that since the resulting height is 50 cm, the 100 cm side must have been rolled to form the circumference. The surface area is calculated as 100 cm × 50 cm = 5000 cm². The most critical step in UPSC CSAT is the unit conversion; because the cost is given per square metre, we must convert our area. Since 1 m² equals 10,000 cm², our area becomes 0.5 m². Multiplying this area by the rate of Rs. 50/m² yields the final result: Rs. 25.00.

UPSC frequently uses distractor options to punish common mistakes. Option (C) Rs. 75.00 is a trap for students who mistakenly calculate the Total Surface Area by attempting to add top and bottom 'lids' to the cylinder, which the problem does not imply. Other incorrect options like (B) and (D) often stem from unit conversion errors (such as dividing by 100 instead of 10,000) or miscalculating the radius and circumference relationship. Always remember: if you are rolling a sheet, the lateral area remains constant regardless of which side you roll it on.