Mensuration Questions for SSC CHSL

Key 2D Shapes: Area and Perimeter

Two-dimensional mensuration deals with flat shapes. You need to know area (space inside) and perimeter (boundary length) for each shape. Rectangle, square, triangle, circle, and trapezium are the most tested in CHSL. The trick is to not confuse similar-looking formulas — for example, area of triangle is half base times height, not just base times height.

• 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; also Area = root of s(s-a)(s-b)(s-c) where s is semi-perimeter
• Circle: Area = pi r squared, Circumference = 2 pi r
• Trapezium: Area = half x (sum of parallel sides) x height
• Equilateral Triangle: Area = (root 3 / 4) x side squared

Key 3D Shapes: Volume and Surface Area

Three-dimensional mensuration deals with solid shapes. Volume is the space inside the object (cubic units) and surface area is the total area of all outer faces. CHSL mostly asks about cylinders, cones, spheres, and cuboids. A common question type: melting one shape and recasting into another — you equate the volumes.

• Cuboid: Volume = l x b x h, Total Surface Area = 2(lb + bh + lh)
• Cube: Volume = side cubed, Total Surface Area = 6 x side squared
• Cylinder: Volume = pi r squared h, Curved Surface Area = 2 pi r h, Total SA = 2 pi r (r + h)
• Cone: Volume = (1/3) pi r squared h, Curved Surface Area = pi r l, where l = slant height = sqrt(r^2 + h^2)
• Sphere: Volume = (4/3) pi r cubed, Surface Area = 4 pi r squared
• Hemisphere: Volume = (2/3) pi r cubed, Curved SA = 2 pi r squared, Total SA = 3 pi r squared

Combined and Conversion Questions

CHSL loves questions where one shape is converted into another (like a wire bent into a circle after being a square), or where a path or border is drawn around a shape. For path/border problems, subtract inner area from outer area. For wire-bending, equate perimeters. For melting and recasting, equate volumes.

• Wire bent into new shape: equate total perimeter or circumference (length of wire stays same)
• Melting and recasting: equate volumes of old and new shapes
• Path around a shape: Area of path = Area of outer boundary minus area of inner boundary
• If a path of width w is around a rectangle l x b, outer dimensions become (l + 2w) x (b + 2w)
• Always check if question asks for curved surface area or total surface area — they are different

Units and Conversions

Unit conversion errors silently kill marks. When area is asked in sq metres but dimensions are in cm, many students forget to convert. Volume conversion is even trickier since it involves cube of the conversion factor.

• 1 m = 100 cm, so 1 sq m = 10000 sq cm, and 1 cubic m = 1000000 cubic cm
• 1 km = 1000 m, 1 hectare = 10000 sq m
• 1 litre = 1000 cubic cm (very common in cylinder/tank problems)
• Always convert all measurements to the same unit before plugging into formula

• Using diameter instead of radius in circle and sphere formulas — always halve the diameter first.
• Forgetting the factor of one-third in cone volume — cone holds exactly one-third of cylinder with same base and height.
• Confusing curved surface area with total surface area for cylinder and cone — CSA does not include the circular base(s).
• Not converting units before calculating — dimensions in cm but answer expected in metres means area answer must be divided by 10000.
• In path problems, adding width once instead of twice to each side — a path around a rectangle adds width to both sides, so each dimension increases by 2w, not w.

• CONE is ONE-THIRD of cylinder: think C-ONE, the C stands for one-third. Cone volume = (1/3) cylinder volume.
• For sphere: Volume has 4-3 in it (4/3), Surface area has just 4 (4 pi r squared). More surface, simpler formula.
• PATH problems: think of adding a frame to a photo. The frame adds width on BOTH left-right and top-bottom. So add 2w always.
• Slant height of cone = hypotenuse of right triangle. Draw a triangle inside cone: base = r, vertical = h, slant = l. Pythagoras gives l = sqrt(r squared + h squared).

• CHSL typically has 3 to 5 mensuration questions per paper. Cylinder and cone appear most frequently among 3D shapes; circle and rectangle dominate 2D.
• One question almost every year involves melting a solid and recasting — just equate volumes and solve for the unknown dimension.
• Questions are usually one or two steps. If your calculation is going beyond 3 steps, re-read the question — you may have misread radius vs diameter or cm vs m.
• Use pi = 22/7 when radius is a multiple of 7. Use pi = 3.14 only if the question explicitly states it or radius involves decimals.
• In the last 2 years CHSL papers have included questions on combinations like a hemisphere placed on top of a cylinder — surface area of such composite solids. Just add CSA of cylinder plus CSA of hemisphere (not full sphere).