Mensuration Questions for CDS

2D Shapes: Triangles

Triangles appear in almost every CDS paper, either directly or as part of a composite figure. The three formulas you must know cold are the base-height formula, Heron's formula for when all three sides are given, and the equilateral triangle formula. CDS loves giving you a triangle inside a circle or rectangle and asking for the shaded area — so combine these formulas with circle and rectangle formulas.

• Area using base and height: half times base times height
• Heron's formula: first find semi-perimeter s = (a+b+c)/2, then Area = sqrt(s(s-a)(s-b)(s-c))
• Equilateral triangle with side a: Area = (sqrt(3)/4) x a^2, Height = (sqrt(3)/2) x a
• Right triangle: Area = (1/2) x product of two legs (the sides forming the right angle)
• Perimeter is always sum of all three sides

2D Shapes: Quadrilaterals and Circles

Rectangle, square, parallelogram, rhombus, trapezium and circle are all tested. CDS often combines two shapes — for example, a circle inscribed in a square, or a trapezium with a semicircle on top. The key skill is breaking the composite figure into known shapes and adding or subtracting areas. Always double-check whether the question asks for area or perimeter — CDS setters deliberately mix these.

• Rectangle: Area = length x breadth; Perimeter = 2(l + b); Diagonal = sqrt(l^2 + b^2)
• Square with side a: Area = a^2; Perimeter = 4a; Diagonal = a*sqrt(2)
• Parallelogram: Area = base x height (not base x slant side)
• Rhombus: Area = (1/2) x d1 x d2 where d1 and d2 are diagonals
• Trapezium: Area = (1/2) x (sum of parallel sides) x height
• Circle with radius r: Area = pi*r^2; Circumference = 2*pi*r; Semicircle area = (pi*r^2)/2

3D Solids: Cube, Cuboid, Cylinder

These three are bread and butter of CDS 3D mensuration. Cube and cuboid questions often involve painting problems (surface area) or filling problems (volume). Cylinder questions typically involve finding how much material the curved surface is made of, or how many litres a tank holds. Remember: Total Surface Area = Curved Surface Area + two circular ends.

• Cube with edge a: Volume = a^3; Total Surface Area = 6a^2; Diagonal = a*sqrt(3)
• Cuboid l x b x h: Volume = lbh; TSA = 2(lb + bh + lh); Diagonal = sqrt(l^2+b^2+h^2)
• Cylinder radius r, height h: Volume = pi*r^2*h; CSA = 2*pi*r*h; TSA = 2*pi*r*(r+h)
• For a hollow cylinder, volume = pi*h*(R^2 - r^2) where R is outer and r is inner radius
• Number of smaller cubes from a big cube = (big side / small side)^3

3D Solids: Cone, Sphere, Hemisphere

These three solids appear in nearly every CDS paper. The slant height of a cone is a common trick — always compute l = sqrt(r^2 + h^2) first if not given. For sphere problems, remember surface area uses r^2 and volume uses r^3. CDS loves conversion questions: melting a sphere and recasting as a cone or cylinder — set volumes equal and solve.

• Cone: Volume = (1/3)*pi*r^2*h; CSA = pi*r*l; TSA = pi*r*(r+l); Slant height l = sqrt(r^2+h^2)
• Sphere radius r: Volume = (4/3)*pi*r^3; Surface Area = 4*pi*r^2
• Hemisphere: Volume = (2/3)*pi*r^3; CSA = 2*pi*r^2; TSA = 3*pi*r^2
• Melting/recasting: equate volumes of old shape and new shape
• If a sphere is cut into two equal halves, each piece TSA = 3*pi*r^2 (curved half + one flat circle)

• Using base x side instead of base x height for parallelogram area — the slant side is NOT the height
• Forgetting to add the two circular ends when TSA of cylinder is asked — CSA and TSA are different and CDS exploits this
• Not computing slant height before calculating cone CSA — plugging h instead of l into pi*r*l gives a wrong answer
• Confusing diameter and radius — CDS question may give diameter; always halve it before plugging into formulas
• Unit mismatch in volume problems — if dimensions are in cm, volume is in cubic cm not litres; 1 litre = 1000 cubic cm

• CAVE for Cone: CSA = pi*r*l, Area-Total = pi*r*(r+l), Volume = (1/3)*pi*r^2*h, Extra = l = sqrt(r^2+h^2)
• Sphere 4 and 3 rule: Surface Area has 4 (4*pi*r^2), Volume has 4/3 (4/3*pi*r^3) — both start with 4
• Rhombus: think of a kite — area is always HALF the product of the two diagonals, like a kite frame divided by 2
• Melting-recasting trick: Volume In = Volume Out — nothing is created or destroyed, just reshape the same material

• CDS typically places 5 to 8 mensuration questions per paper; at 2 marks each this is 10 to 16 marks — a highly rewarding topic to master
• Recent papers favour combination shape problems (e.g. cone on top of cylinder, or circle inside square) — practice these composite figures specifically
• Recasting and melting questions (sphere to cylinder, cylinder to cone) appear almost every year — they always reduce to setting volumes equal
• Use pi = 22/7 unless the question explicitly says use 3.14; most CDS options are designed around 22/7
• Time-saving tip: for MCQs, back-calculate from answer options instead of solving fully — plug options into the formula to see which satisfies the equation, especially when algebra gets messy