Geometry and Mensuration for CSAT: PYQs Explained

Mensuration is the branch of mathematics that measures lengths, areas and volumes of geometric figures. In CSAT Paper II, which is a qualifying paper (roughly 33 percent clears it; the marks do not add to your merit rank), UPSC rarely asks for long, grinding area calculations. Instead it tests three skills: counting figures hidden in lines and points, tiling and fitting shapes inside other shapes, and reasoning about rolling and rotating objects. One page of formulae plus these three patterns covers nearly every geometry question the paper has asked.

The formulae you actually need

You need the area and perimeter of the flat (two-dimensional) shapes, and the volume and surface area of the solid (three-dimensional) ones. Memorise these once; every worked example below uses them.

Pattern 1: counting figures from lines and points

A favourite UPSC pattern gives you a grid of parallel lines and asks how many rectangles it contains. The rule: a rectangle is fixed by choosing 2 of the horizontal lines and 2 of the vertical lines. With m horizontal and n vertical lines, the count is C(m, 2) times C(n, 2), where C(m, 2) = m(m minus 1)/2. Remember that squares are a special kind of rectangle, so they are already included in this count unless the question says otherwise.

Worked example (UPSC CSE 2017). 4 horizontal and 4 vertical lines are drawn on a board. How many rectangles and squares can be formed? Choose 2 horizontals: C(4, 2) = 6. Choose 2 verticals: C(4, 2) = 6. Rectangles = 6 times 6 = 36, and this already includes the squares (nine 1-by-1, four 2-by-2, one 3-by-3). The answer is option (c), 36.

Worked example (UPSC CSE 2018). 24 equally spaced points lie on the circumference of a circle. What is the maximum number of equilateral triangles with these points as vertices? An equilateral triangle on a circle needs its vertices to subtend 120 degrees at the centre, i.e. every 8th point. Starting points 1 through 8 give 8 distinct triangles (starting at point 9 repeats the one from point 1), so the answer is 8, option (c). The general idea: symmetric figures are counted by how many distinct starting positions survive the symmetry.

Pattern 2: tiling and fitting shapes

Here the stem asks whether one shape can be cut or fitted into another. The two-step method is: first check area conservation (the pieces cannot exceed the whole), then check geometric fitting (the side lengths must actually divide the sides). Area passing while fitting fails is the standard trap.

Worked example (UPSC CSE 2018). 12 equal squares are placed without gaps to form a rectangle of diagonal 5 cm, in 3 rows of 4 squares. What is the area of each square? Let each square have side s. The rectangle is 4s by 3s, so its diagonal is 5s by the 3-4-5 triangle. But the diagonal is given as 5 cm, so 5s = 5, giving s = 1 cm and each square an area of 1 square cm, option (c).

Worked example (UPSC CSE 2022). A 20 cm by 8 cm rectangular sheet: (1) can it be cut exactly into 4 square sheets? (2) can it be cut into 10 triangular sheets of equal area? Statement 1: four equal squares would each have area 160/4 = 40 square cm, side root 40, which is irrational and cannot tile the 20-by-8 rectangle exactly, so statement 1 is false. Statement 2: ten triangles of equal area need only 16 square cm each, and triangles of any shape with that area can be cut from the sheet, so statement 2 is true. The answer is 2 only, option (b).

Pattern 3: rolling and circumference puzzles

When a wheel rolls without slipping, the number of rotations equals the distance travelled divided by the wheel's own circumference: rotations = distance / (2 pi r). The pi cancels whenever two wheels are compared, which is exactly what UPSC exploits.

Worked example (UPSC CAPF 2016). Two concentric circles have radii 100 m and 110 m. A wheel of radius 30 cm rolls on the smaller circle and another wheel on the larger circle; both complete one revolution of their circles with the same number of rotations. What is the radius of the other wheel? Rotations on the small circle = 2 pi times 100 m divided by 2 pi times 0.30 m = 100/0.30. Setting this equal to 110/r for the big circle gives r = 30 times 110/100 = 33 cm, option (c). The pi and the 2 cancel on both sides, so you only need the ratio of the radii.

Traps UPSC sets

• Area versus perimeter. If the stem asks for the fencing needed, it wants a perimeter; if it asks for the carpet, it wants an area. Re-read the noun in the question before computing.
• Radius versus diameter. Circle formulae use the radius; a stem that gives a diameter must be halved first. The most expensive single error in mensuration.
• Squares hide inside rectangle counts. A square is a rectangle. When the question asks for rectangles, the formula C(m,2) times C(n,2) already counts the squares; only exclude them if the stem says so.
• Units must match. Convert metres to centimetres (or vice versa) before dividing; the wheel puzzle above fails instantly if 100 m is not converted against 30 cm.
• Area passing is not enough. Four squares of area 40 square cm fit the 20-by-8 sheet by area, but their irrational side length cannot tile it; always run the fitting check after the area check.

Speed tactics

• Memorise the small combinations. C(4,2) = 6, C(5,2) = 10, C(6,2) = 15: rectangle-counting questions then finish in one line.
• Use the 3-4-5 triangle. A rectangle of sides 4s and 3s has diagonal 5s with no calculation; spot the multiple of 3-4-5 the moment a diagonal appears.
• Let pi cancel. In rolling and circumference comparisons, work with ratios of radii and never multiply out pi.
• Check area first, fit second. If the total piece area exceeds the sheet, the statement is false in seconds; only when area passes do you test the side lengths.

Key Terms

• Area: the amount of flat surface a two-dimensional figure covers, measured in square units.
• Perimeter: the total distance around a two-dimensional figure; the length of its boundary.
• Circumference: the perimeter of a circle, equal to 2 pi times the radius.
• Radius: the distance from the centre of a circle to any point on it; half the diameter.
• Diameter: the longest straight line through a circle's centre, equal to twice the radius.
• Diagonal: a straight line joining two non-adjacent corners of a polygon, such as the 5s diagonal of a 4s-by-3s rectangle.
• Polygon: a closed flat figure with straight sides, such as triangles, rectangles and pentagons.
• Regular polygon: a polygon whose sides and angles are all equal, such as a square or an equilateral triangle.
• Chord: a straight line joining two points on a circle's circumference; the diameter is the longest chord.
• Volume: the amount of space a three-dimensional solid occupies, measured in cubic units.
• Surface area: the total area of all faces of a solid, including the base; for a hemisphere the total is 3 pi r squared.
• Lateral (curved) surface area: the area of the sides of a solid excluding its base(s), such as 2 pi r h for a cylinder.

Practice questions