Mensuration Questions for SBI PO

Key 2D Shapes: Area and Perimeter

Two-dimensional mensuration covers flat shapes. You need both area (space inside) and perimeter (boundary length). SBI PO frequently combines two shapes — like a rectangle with semicircles on its ends — so treat each sub-shape separately and add up.

• Rectangle: Area = l x b, Perimeter = 2(l + b)
• Square: Area = side squared, Perimeter = 4 x side, Diagonal = side x root(2)
• Triangle: Area = 0.5 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 = 0.5 x (sum of parallel sides) x height
• Rhombus: Area = 0.5 x d1 x d2 (product of diagonals halved)

Key 3D Shapes: Volume and Surface Area

Three-dimensional mensuration tests volume (space inside) and surface area (total outer skin). SBI PO loves cylinder and cone problems, often asking you to compare volumes when dimensions change. A common trick: if radius doubles, volume becomes 4 times (because r is squared in cylinder formula).

• Cube: Volume = a cubed, Total Surface Area (TSA) = 6 x a squared, Diagonal = a x root(3)
• Cuboid: Volume = l x b x h, TSA = 2(lb + bh + lh)
• 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 = (1/3) x pi x r squared x h, CSA = pi x r x l, where slant l = sqrt(r squared + h squared)
• Sphere: Volume = (4/3) x pi x r cubed, Surface Area = 4 x pi x r squared
• Hemisphere: Volume = (2/3) x pi x r cubed, CSA = 2 x pi x r squared, TSA = 3 x pi x r squared

Combined and Conversion Problems

SBI PO especially favours problems where a shape is melted or converted into another — a sphere melted into small spheres, or water flowing through a pipe filling a tank. The key principle: volume stays constant in melting or filling problems. Set up an equation equating volumes.

• Melting/recasting: Volume of original shape = total volume of new shapes
• Water flow: Volume of water = cross-section area of pipe x speed x time
• Painting/fencing problems: calculate only the relevant surface (TSA vs CSA vs base excluded)
• Cost problems: multiply area or volume by rate per unit
• If n small spheres are made from one big sphere: n x (4/3 pi r-small cubed) = (4/3 pi R-big cubed)

Ratio and Scaling Shortcuts

When dimensions change by a factor, areas and volumes change by the square or cube of that factor. SBI PO loves these shortcut questions because aspirants who do not know the scaling rules waste time re-calculating from scratch.

• If side of square doubles: area becomes 4 times (2 squared)
• If radius of circle doubles: area becomes 4 times
• If all dimensions of cuboid double: volume becomes 8 times (2 cubed)
• If radius of sphere triples: volume becomes 27 times (3 cubed), surface area becomes 9 times
• Percentage increase in area when side increases by x percent: use (100 + x) squared / 100 squared - 1

• Using diameter instead of radius in circle formulas — always halve the diameter before plugging into pi x r squared
• Forgetting to use slant height (not vertical height) when calculating cone CSA — slant l = sqrt(r squared + h squared) must be computed first
• Confusing CSA and TSA for cylinders and hemispheres — in open tank problems use only CSA plus base; read the question carefully about what surface is being painted or covered
• In melting problems, equating surface areas instead of volumes — volume is conserved in melting, not surface area
• Not converting units before calculating — mixing metres and centimetres leads to wrong answers in pipe and tank problems

• Cone = Cylinder divided by 3 — a cone always holds exactly one-third of a cylinder with the same base and height. Visualise pouring a cone three times to fill the cylinder.
• Acronym SCCT for sphere surface areas: S = 4 pi r squared (sphere), C = 3 pi r squared (closed hemisphere TSA = 3), C = 2 pi r squared (curved hemisphere only = 2). Numbers go 4, 3, 2.
• For scaling: AREA squares, VOLUME cubes — if something is scaled by k, area goes k-squared, volume goes k-cubed. Think: 2D needs 2 multiplications, 3D needs 3.
• Diagonal memory: Square diagonal = side x root(2), Cuboid diagonal = root(l squared + b squared + h squared), Cube diagonal = side x root(3). Dimensions keep adding under the root.

• SBI PO Prelims typically has 2-3 direct mensuration questions; Mains DI sets often embed volume or area calculations inside data tables, so formula recall must be instant.
• Expect at least one combined-shape or melting-and-recasting problem in Mains — these take 2-3 minutes if you set up the equation correctly from the start, but 5+ minutes if you go by trial.
• SBI PO sets questions where a cylinder is open at the top (exclude one base from TSA) or a hollow pipe (subtract inner cylinder volume from outer) — read every word of the problem.
• Use pi = 22/7 when radius is a multiple of 7 (7, 14, 21); use pi = 3.14 otherwise. Mixing these is a time-waster.
• If a question gives cost of painting per sq metre, immediately calculate only the surface relevant (walls of room = CSA of four walls, floor excluded unless stated) — do not auto-compute full TSA.