Mensuration Questions for RRB NTPC

2D Shapes: Area and Perimeter

Two-dimensional mensuration deals with flat shapes. You need to memorize area and perimeter formulas cold — there is no time to derive them in the exam. The most tested shapes are rectangle, square, triangle, circle, and trapezium. Watch out for questions that give perimeter and ask for area, or vice versa — these require two steps.

• Rectangle: Area = length x breadth; Perimeter = 2(l + b)
• Square: Area = side squared; Perimeter = 4 x side; Diagonal = side x root 2
• Triangle: Area = half x base x height; for equilateral = (root3 / 4) x side squared
• Circle: Area = pi x r squared; Circumference = 2 x pi x r
• Trapezium: Area = half x (sum of parallel sides) x height
• Rhombus: Area = half x product of diagonals

3D Shapes: Volume and Surface Area

Three-dimensional mensuration covers solids. RRB NTPC loves cylinder and cuboid questions — often framed as tanks, pipes, or boxes. Key distinction: Total Surface Area (TSA) includes all faces; Curved or Lateral Surface Area (CSA/LSA) excludes the base and top. Volume questions often involve cost of filling or capacity problems.

• Cuboid: Volume = l x b x h; TSA = 2(lb + bh + lh)
• Cube: Volume = a cubed; TSA = 6 x a squared; Diagonal = a x root 3
• Cylinder: Volume = pi x r squared x h; CSA = 2 x pi x r x h; TSA = 2 x pi x r x (r + h)
• Cone: Volume = one-third x pi x r squared x h; CSA = pi x r x l; where slant height l = root(r squared + h squared)
• Sphere: Volume = four-thirds x pi x r cubed; TSA = 4 x pi x r squared
• Hemisphere: Volume = two-thirds x pi x r cubed; CSA = 2 x pi x r squared; TSA = 3 x pi x r squared

Unit Conversions and Scaling

Many NTPC mistakes happen not in formula recall but in unit conversion. Area scales as the square of the linear dimension; volume scales as the cube. If a side doubles, area becomes 4 times, volume becomes 8 times. Questions on 'carpet cost' require converting sq metres to sq cm or vice versa carefully.

• 1 metre = 100 cm; 1 sq metre = 10000 sq cm; 1 cubic metre = 1000000 cubic cm
• 1 hectare = 10000 sq metres — used in land area questions
• If radius doubles: Area becomes 4 times, Volume becomes 8 times
• If each side of cube is tripled: Volume becomes 27 times
• Always check whether diameter or radius is given for circle/cylinder/sphere problems

Combined and Composite Shapes

RRB NTPC occasionally asks about shapes made by combining two figures — for example, a cylinder topped with a cone (like a rocket), or a rectangular field with a semicircular end. Strategy: break into known shapes, find each part separately, then add. For hollow cylinders or rings, subtract the inner from the outer.

• Composite shape area or volume = sum of individual parts
• Ring (annulus) area = pi x (R squared minus r squared)
• Hollow cylinder volume = pi x (R squared minus r squared) x h
• Read the problem carefully to know if both ends are open or closed when finding CSA vs TSA
• For path-around problems: outer area minus inner area gives path area

• Using diameter instead of radius in circle, cylinder, sphere formulas — always halve the diameter first before plugging in.
• Confusing CSA and TSA for cylinder and cone — if the question says 'sheet needed to make a can with open top', use CSA + one base only.
• Forgetting to find slant height before computing cone CSA — you must calculate l = root(r squared + h squared) first.
• Unit mismatch — mixing metres and centimetres in cost-of-tiling problems leads to answers off by factors of 10000.
• In scaling questions, applying the factor once instead of squaring it for area or cubing it for volume — e.g. if side doubles, area is 4 times NOT 2 times.

• Cylinder, Cone, Sphere — CCS: all share pi and r. Cylinder is full, Cone is one-third, Sphere has that 4/3 with r cubed. Remember: C-Full, Co-Third, S-Four-Third.
• For TSA of hemisphere remember 3: TSA = 3 pi r squared. Think of it as 2 (curved) + 1 (flat base) = 3 parts of pi r squared.
• PACT mnemonic for 2D shapes: Parallelogram = base x height; Area of circle = pi r squared; Circle Circumference = 2 pi r; Trapezium = half sum of parallels x height.
• Scaling trick: Area goes up by the SQUARE, Volume goes up by the CUBE. Remember: 2D needs square, 3D needs cube — same as their dimension count.

• RRB NTPC Stage 1 typically has 3 to 4 mensuration questions out of 30 Quant questions. Stage 2 may have slightly more with higher difficulty.
• Circle and cylinder are the most repeated shapes — prioritize these two above all others. Sphere and cone appear occasionally.
• Questions are often indirect: 'cost of painting', 'volume of water to fill a tank', 'area of a path around a garden' — learn to identify which formula applies from the real-world context.
• Use pi = 22/7 when radius is a multiple of 7, and pi = 3.14 otherwise — this avoids messy decimals and speeds up calculation.
• Target time: 60 to 90 seconds per mensuration question. If you cannot recall the formula in 10 seconds, mark for review and move on — do not waste time deriving formulas during the exam.