LMNOP, is a semi-circle with centre at R and diameter LP, LSR and RQP are also semi-circles with centres at T and U and diameters LR = RP = 1/2 LP. The ratio of perimeters of LMNOP and LSRQP is

1. Circle Geometry Fundamentals (basic)

To master geometry, we must begin with the most perfect shape: the circle. At its simplest, a circle is a collection of all points in a plane that are at a fixed distance, called the radius (r), from a central point. When we talk about the distance across the circle passing through the center, we are referring to the diameter (d). A fundamental rule you must internalize is that the diameter is always twice the radius (d = 2r), a relationship similarly reflected in optics where the radius of curvature of a mirror is twice its focal length Science, Class X, p.137.

When we measure the "boundary" of a circle, we call it the circumference. The formula for this is 2πr (or πd). However, things get interesting when we look at a semicircle (half a circle). A common mistake in competitive exams is assuming the perimeter of a semicircle is simply half the circumference. In reality, a semicircle's perimeter consists of two parts: the curved arc (πr) and the flat base, which is the diameter (2r). Therefore, Perimeter of a Semicircle = πr + 2r.

Property	Full Circle	Semicircle
Curved Boundary	2πr	πr
Total Perimeter	2πr	πr + 2r (or πr + d)

In geography, we apply these circular principles to the Earth. While the Earth is a sphere, we represent its cross-sections as circles. The largest possible circles that can be drawn on a sphere are Great Circles, such as the Equator Certificate Physical and Human Geography, The Earth's Crust, p.14. Understanding how these circles scale is vital: if you halve the diameter of a circle, you also halve its circumference. This linear relationship is the secret to solving complex ratio problems involving multiple circular shapes inscribed within one another.

Sources: Science, Class X, Light – Reflection and Refraction, p.137; Certificate Physical and Human Geography, The Earth's Crust, p.14

2. Arc Length and Sector Calculations (intermediate)

To understand the geometry of curved paths, we begin with the fundamental property of a circle: it represents a full rotation of 360°. As noted in geography, the Earth's rotation of 360° every 24 hours allows us to divide the globe into meridians Exploring Society: India and Beyond, Class VI, Locating Places on the Earth, p.20. When we talk about the Arc Length, we are simply measuring a portion of the circle's total boundary (the circumference, 2πr). If a full circle is 360°, then an arc spanning an angle of θ is calculated as (θ/360) × 2πr. In physical geography, the shortest distance between two points on a sphere lies along a Great Circle, which is the largest possible circle that can be drawn on a sphere, such as the Equator Certificate Physical and Human Geography, GC Leong, The Earth's Crust, p.14. Moving from arcs to Sectors, think of a sector as a 'pizza slice' defined by two radii and an arc. A common point of confusion in competitive exams is the difference between arc length and perimeter. While the arc length is just the curved edge, the Perimeter of a Sector must include the two straight radii that 'close' the shape. Therefore, Perimeter = [(θ/360) × 2πr] + 2r. This logic is critical when dealing with Semicircles (where θ = 180°). The perimeter of a semicircle is half the circumference plus the diameter: πr + 2r. When solving intermediate problems, always look for the relationship between the Radius (r) and the Diameter (d = 2r). Just as in spherical mirrors where the focal length relates to the radius of curvature Science, Class X, Light – Reflection and Refraction, p.143, in arc calculations, doubling the radius will double both the arc length and the total perimeter if the angle remains constant. Understanding these linear relationships allows you to compare different shapes—like a large semicircle versus two smaller ones—by expressing all dimensions in terms of a single variable.

Sources: Exploring Society: India and Beyond, Class VI, Locating Places on the Earth, p.20; Certificate Physical and Human Geography, GC Leong, The Earth's Crust, p.14; Science, Class X, Light – Reflection and Refraction, p.143

3. Mensuration of Semi-circles (basic)

To understand a semi-circle, we must first visualize a full circle cut exactly in half along its diameter. While a circle is a continuous curved boundary, a semi-circle is a closed shape consisting of one curved arc and one straight diameter. This distinction is crucial for aptitude exams because the straight edge is often forgotten in calculations. Just as the Earth's 'great circles' represent the largest possible circumferences around its spherical shape Certificate Physical and Human Geography, GC Leong (Oxford University press 3rd ed.), The Earth's Crust, p.14, a semi-circle represents half of that 2D circular boundary.

The Area of a semi-circle is straightforward: it is exactly half the area of a full circle. Since the area of a circle is πr², the area of a semi-circle is ½πr². However, calculating the Perimeter (or circumference) requires more care. If you bend a straight wire into a circular loop, the length is 2πr Science, class X (NCERT 2025 ed.), Magnetic Effects of Electric Current, p.200. But for a semi-circle, the perimeter includes the curved half-arc (πr) PLUS the straight base that closes the shape, which is the diameter (d or 2r).

Feature	Circle Formula	Semi-circle Formula
Area	πr²	½ πr²
Perimeter	2πr	πr + 2r (or r[π + 2])

Sources: Certificate Physical and Human Geography, GC Leong (Oxford University press 3rd ed.), The Earth's Crust, p.14; Science, class X (NCERT 2025 ed.), Magnetic Effects of Electric Current, p.200

4. Principles of Scaling in Geometry (intermediate)

In geometry, Scaling refers to the process of increasing or decreasing the size of a figure while maintaining its original proportions. When we scale a geometric figure, all of its linear dimensions—such as radius, diameter, side length, or height—change by the same constant factor, often called the scale factor (k). A fundamental principle to master for aptitude tests is that the perimeter or circumference of a shape scales linearly. This means if you double the radius of a circle, you exactly double its circumference. This relationship is often seen in physics and geography as well; for instance, the radius of a mirror is exactly twice its focal length (R = 2f), representing a direct 1:2 linear ratio Science, class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.137.
When dealing with semicircles, the perimeter is composed of two parts: the curved arc (πr) and the straight-line diameter (2r). Thus, Perimeter = r(π + 2). If we reduce the radius by half, the new perimeter becomes (r/2)(π + 2), which is exactly half of the original. An interesting consequence of this linear scaling is the Additive Property of Perimeters. If a large line segment is divided into several smaller segments, and a semicircle is constructed on each segment, the sum of the perimeters of those small semicircles will exactly equal the perimeter of a single large semicircle constructed on the original line. This is because the sum of the smaller diameters equals the large diameter, and due to linear scaling, the sum of the smaller arcs equals the large arc.
In a global context, we see these principles applied to the Earth's geometry. The shortest distance between two points on a sphere lies along a Great Circle, which is the largest possible circle that can be drawn on a sphere, representing the maximum circumference Certificate Physical and Human Geography, GC Leong (Oxford University press 3rd ed.), The Earth's Crust, p.14. Because the Earth is an oblate spheroid rather than a perfect sphere, its radius is larger at the equator than at the poles Physical Geography by PMF IAS, Manjunath Thamminidi, PMF IAS (1st ed.), Latitudes and Longitudes, p.241. This variation in radius directly affects the calculated 'perimeter' or circumference at different latitudes, reinforcing that in geometry, the total boundary length is always a direct function of the radius.

Sources: Science, class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.137; Certificate Physical and Human Geography, GC Leong (Oxford University press 3rd ed.), The Earth's Crust, p.14; Physical Geography by PMF IAS, Manjunath Thamminidi, PMF IAS (1st ed.), Latitudes and Longitudes, p.241

5. Ratio and Proportion in Mensuration (exam-level)

In the world of quantitative aptitude, Ratio and Proportion act as the bridge between raw measurements and meaningful comparisons. Just as we use ratios to compare social indicators like the Child Sex Ratio across different districts Geography of India, Majid Husain, Cultural Setting, p.84, we use them in mensuration to understand how changes in a figure's dimensions affect its overall properties. When we deal with curved surfaces or shapes, such as the curved mirrors we see in science Science, Class VIII, Light: Mirrors and Lenses, p.162, the ratio of their radii determines how the entire structure scales.

Let’s look at the Perimeter of a Semicircle. It is composed of two parts: the curved arc (half the circumference) and the straight diameter. Mathematically, if the radius is r, the perimeter is πr + 2r, or r(π + 2). Notice that the perimeter is linearly proportional to the radius. This means if you halve the radius, the perimeter also halves. If you have two such smaller semicircles, their combined perimeters will exactly equal the perimeter of the original larger semicircle, because 1/2 + 1/2 = 1. This linear relationship is a fundamental shortcut in exam problems; you don't always need to calculate the value of π to find the ratio.

This concept of proportionality is everywhere. Whether we are analyzing the Head Count Ratio of poverty in different states Economics, Class IX, Poverty as a Challenge, p.34 or comparing the shapes of different human cells Science, Class VIII, The Invisible Living World: Beyond Our Naked Eye, p.13, we are essentially looking at how one part relates to the whole. In mensuration, remember that linear dimensions (like perimeter or height) scale at the same rate as the radius, whereas area scales with the square of the radius. Mastering this distinction allows you to solve complex geometry problems by simply observing the ratios of their base dimensions.

Sources: Geography of India, Majid Husain, Cultural Setting, p.84; Science, Class VIII, Light: Mirrors and Lenses, p.162; Economics, Class IX, Poverty as a Challenge, p.34; Science, Class VIII, The Invisible Living World: Beyond Our Naked Eye, p.13

6. Area and Perimeter of Composite Regions (exam-level)

In the realm of quantitative aptitude, a composite region is a figure constructed by combining two or more basic geometric shapes, such as rectangles, triangles, or circles. To master these, we must look beyond the individual formulas and understand how boundaries interact. For instance, when we study the Earth's orbit as an oval-shaped path or represent it using concentric circles of varying radii Science-Class VII, Earth, Moon, and the Sun, p.186, we are essentially looking at the relationship between different geometric layers. The most critical rule for Perimeter in composite regions is that we only count the outermost boundary; any line segment where two shapes touch becomes internal and is excluded from the total perimeter calculation.
When dealing with curved composite shapes, such as semicircles, we must be precise with our components. The perimeter of a standalone semicircle consists of two parts: the curved arc (half the circumference, or πr) and the straight diameter (2r). Interestingly, if you take a large semicircle and replace its straight diameter with several smaller semicircles whose diameters add up to the original length, a unique symmetry emerges. Even though the area might change significantly, the total length of the curved boundaries remains constant because the sum of the smaller radii always equals the large radius. This principle is often used in design and mapping to represent divided geographical areas Physical Geography by PMF IAS, Pressure Systems and Wind System, p.305.
For Area, the approach is additive or subtractive. You simply calculate the area of each component part and sum them up. Unlike perimeter, internal boundaries do not disappear in area calculations; they define the limits of each sub-section. If a problem asks you to find the area of a shaded region (a common UPSC favorite), you often subtract the area of an 'internal' shape from a 'host' shape. This requires a sharp eye for palaeography-like precision THEMES IN INDIAN HISTORY PART I, Kings, Farmers and Towns, p.29—noticing exactly where one shape ends and another begins to avoid double-counting overlapping regions.

Sources: Science-Class VII, Earth, Moon, and the Sun, p.186; Physical Geography by PMF IAS, Pressure Systems and Wind System, p.305; THEMES IN INDIAN HISTORY PART I, Kings, Farmers and Towns, p.29

7. Boundary Calculation for Multiple Semi-circles (intermediate)

To calculate the boundary of a semicircle, we must account for two distinct components: the curved arc length and the straight diameter (base). Just as the Earth's circumference represents the path of a "Great Circle" Certificate Physical and Human Geography, The Earth's Crust, p.14, the arc of a semicircle is exactly half of a full circle's circumference (πr). Therefore, the full boundary, or perimeter, of a closed semicircle is defined by the formula: Perimeter = πr + 2r (where r is the radius) or (π/2)D + D (where D is the diameter).

When a large semicircle is subdivided into multiple smaller semicircles whose diameters lie along the original diameter's line, a significant property of proportionality arises. If you draw circles with different radii Science-Class VII, Earth, Moon, and the Sun, p.186, you will observe that both the circumference and the diameter scale linearly. If a large diameter is split into n smaller segments, each small semicircle's arc and base are reduced by that same factor. Consequently, the sum of the boundaries of these small semicircles will always be exactly equal to the boundary of the original large semicircle.

This principle of geometric scaling is universal, whether applied to the tiny diameters of soil particles Geography of India, Soils, p.2 or the geometric layouts of rectangular settlements Geography of India, Settlements, p.6. Because the ratio of the arc to the diameter remains constant (π/2), any linear combination of smaller semicircles that completes a larger base will preserve the total perimeter length.

Feature	Large Semicircle (Radius R)	n Small Semicircles (Radius r = R/n)
Curved Arc	πR	n × π(R/n) = πR
Straight Base	2R	n × 2(R/n) = 2R
Total Boundary	R(π + 2)	R(π + 2)

Sources: Certificate Physical and Human Geography, The Earth's Crust, p.14; Science-Class VII, Earth, Moon, and the Sun, p.186; Geography of India, Soils, p.2; Geography of India, Settlements, p.6

8. Invariant Properties in Circle Geometry (exam-level)

To master geometry for competitive exams, one must understand scaling and invariant properties. When we deal with circles and semicircles, the relationships between radius, diameter, and perimeter are linear. A semicircle's perimeter consists of two parts: the curved arc (πr) and the straight base or diameter (2r). As noted in spherical studies, the radius of curvature is precisely twice the focal length (R = 2f), illustrating the fixed 2:1 ratio in circular dimensions Science, Light – Reflection and Refraction, p.137.

Consider a large semicircle with diameter D. Its total perimeter is P = π(D/2) + D. If we divide this diameter into n smaller segments that sum up to D, and build smaller semicircles on each, a fascinating property emerges: The sum of the perimeters of the smaller semicircles will always equal the perimeter of the larger one. This is because both the curved arcs and the straight bases scale linearly. For instance, if you have two small semicircles with diameters d₁ and d₂ such that d₁ + d₂ = D, their combined perimeter is [π(d₁/2) + d₁] + [π(d₂/2) + d₂]. Factoring this out gives π/2(d₁ + d₂) + (d₁ + d₂), which simplifies back to π(D/2) + D.

This concept of "equal lengths" is mirrored in physical geography; for example, all Great Circles on a sphere (like the Equator or a circle formed by opposite meridians) have the same circumference because they share the same maximum diameter of the sphere Certificate Physical and Human Geography, The Earth's Crust, p.14. In quantitative aptitude, recognizing this 1:1 ratio allows you to bypass complex calculations. Whether you divide a diameter into two, three, or infinite parts, as long as the small shapes are similar and their bases lie along the original diameter, the total distance remains invariant.

Component	Large Semicircle (Diameter D)	Combined Small Semicircles (Σd = D)
Curved Length	πD / 2	π(d₁ + d₂ + ... + dₙ) / 2 = πD / 2
Straight Base	D	d₁ + d₂ + ... + dₙ = D
Total Perimeter	D(π/2 + 1)	D(π/2 + 1)

Sources: Science (NCERT 2025), Light – Reflection and Refraction, p.137; Certificate Physical and Human Geography (GC Leong), The Earth's Crust, p.14

9. Solving the Original PYQ (exam-level)

This problem beautifully integrates your knowledge of Circle Geometry with the ability to visualize composite figures. To solve this, you must apply the fundamental formula for the Perimeter of a Semicircle, which is the sum of the curved arc ($πr$) and the straight diameter ($2r$). By breaking the large semicircle $LMNOP$ and the composite shape $LSRQP$ into these two distinct components, you can see how the linear relationship between radius and circumference plays out in practice.

Let's walk through the logic: if we let the diameter of the large semicircle be $D$, its perimeter is simply $\frac{πD}{2} + D$. Now, look at the figure $LSRQP$. It consists of two smaller semicircular arcs and the same base diameter $LP$. Since the diameter of each small semicircle is exactly $\frac{D}{2}$, the sum of their two curved arcs is $\frac{π(D/2)}{2} + \frac{π(D/2)}{2} = \frac{πD}{2}$. When you add the common base diameter $D$ to these arcs, the total perimeter of $LSRQP$ becomes identical to $LMNOP$. Therefore, the ratio of their total boundaries is exactly 1 : 1.

UPSC often uses such visual puzzles to lure students into common traps. Option (A) and (C) are designed for those who might intuitively feel that smaller internal shapes must have a smaller perimeter, or those who incorrectly apply the area formula (where the ratio would be different). A frequent mistake is forgetting to include the diameter in the perimeter calculation or misidentifying the path of $LSRQP$. As noted in NCERT Mathematics Class X, always remember that while area scales quadratically, the perimeter (a linear measure) scales proportionally with the radius, leading to this 1 : 1 symmetry.