Mensuration and Statistics Questions for CDS

2D Mensuration: Triangles, Quadrilaterals, Circles

These are the bread-and-butter shapes. CDS frequently tests area of a triangle using Heron's formula when all three sides are given, and the standard half-base-height formula otherwise. Circles appear in problems involving sectors, arcs, and combinations with squares or rectangles. Always confirm whether diameter or radius is given — this single slip wastes marks.

• Area of triangle = half x base x height; for sides a,b,c use Heron's formula
• Area of circle = pi r squared; Circumference = 2 pi r
• Area of sector = (theta/360) x pi r squared; Arc length = (theta/360) x 2 pi r
• Area of trapezium = half x (sum of parallel sides) x height
• Area of rhombus = half x product of diagonals
• Area of equilateral triangle = (root 3 / 4) x side squared

3D Mensuration: Cylinder, Cone, Sphere, Cube, Cuboid

Volume and surface area of 3D shapes are tested heavily. CDS loves problems where a shape is melted or reshaped into another — set the volumes equal and solve. Lateral surface area versus total surface area is a common distinction. Hollow cylinders and frustum (truncated cone) appear occasionally in harder sets.

• Cube: Volume = a cubed; Total SA = 6a squared; Diagonal = a root 3
• Cuboid: Volume = l x b x h; Total SA = 2(lb + bh + hl)
• Cylinder: Volume = pi r squared h; Curved SA = 2 pi r h; Total SA = 2 pi r (r + h)
• Cone: Volume = one-third pi r squared h; Curved SA = pi r l; l = sqrt(r squared + h squared)
• Sphere: Volume = (4/3) pi r cubed; SA = 4 pi r squared
• Hemisphere: Volume = (2/3) pi r cubed; Total SA = 3 pi r squared

Statistics: Mean, Median, Mode

CDS tests all three measures of central tendency. Mean is the arithmetic average. Median is the middle value when data is arranged in order — for even count, average the two middle values. Mode is the most frequently occurring value. CDS also asks about effect on mean when a value is added, removed, or changed — use the total sum approach, not recalculation.

• Mean = Sum of all values divided by number of values
• For grouped data, Mean = Sum(f x x) / Sum(f), where f is frequency and x is midpoint
• Median: arrange data in order; if n is odd, median = ((n+1)/2)th term; if n is even, average of (n/2)th and (n/2 + 1)th terms
• Mode = value that appears most often; a dataset can be bimodal
• Range = Maximum value minus Minimum value
• If one incorrect value is replaced, new sum = old sum - wrong value + correct value; recalculate mean

Statistics: Variance and Standard Deviation (Basic)

CDS occasionally asks variance or standard deviation. These measure how spread out data is. Variance is the average of squared deviations from the mean. Standard Deviation (SD) is the square root of variance. For CDS, you rarely need to compute full SD — usually a conceptual question or a small dataset is given.

• Variance = Sum of (xi - mean) squared, divided by n
• Standard Deviation = square root of Variance
• If all values are equal, SD = 0 (no spread)
• Adding a constant to all values does not change SD; multiplying all values by k multiplies SD by k
• CDS rarely asks full computation of SD — focus on properties

• Using diameter instead of radius in circle/cylinder/sphere formulas — always halve the diameter first before substituting
• Forgetting to compute slant height before finding curved surface area of a cone — l = sqrt(r squared + h squared) is a mandatory intermediate step
• In median calculation, not sorting the data before finding the middle position — unsorted data gives completely wrong answer
• Confusing Total Surface Area with Lateral or Curved Surface Area — for cylinder TSA includes two circular ends; CSA does not
• In melting/recasting problems, equating surface areas instead of volumes — always equate volumes unless the question explicitly asks about surface

• CAVE for 3D volumes: Cube = a3, A cylinder = pi r2 h, Volume cone = one-third of cylinder, Everything sphere = four-thirds pi r3
• For median with even n, think PAIR and AVERAGE — pair the two middle ones and average them
• Sector is a SLICE of pizza: both area and arc length use the same fraction theta/360 of the whole circle
• SD properties: Add a constant, SD STAYS; Multiply by k, SD SCALES — remember Add-Stay, Multiply-Scale

• CDS papers typically carry 4-6 Mensuration questions and 3-4 Statistics questions. Together they are among the highest yield areas — do not skip either.
• Mensuration questions are mostly direct formula application at medium difficulty. If a question takes more than 90 seconds, skip and return — it usually means a simpler approach exists.
• Statistics questions in CDS are almost always easy to moderate. Mean and median questions are the most common. A question on correcting a wrong entry using the sum method appears almost every year.
• For 3D problems, always write down what is given (radius, height, slant height) and what is asked (volume, CSA, TSA) before starting — this prevents applying the wrong formula.
• Unit conversion is a hidden trap: if radius is in cm and height is in m, convert before computing. Volume answers may be asked in litres (1 litre = 1000 cubic cm).