Mensuration 3D Questions for SSC CGL

Cube and Cuboid

A cube has all sides equal (side = a). A cuboid has length l, breadth b, height h. These are the simplest 3D shapes and often appear in questions about painting boxes, filling rooms, or cutting shapes. The diagonal of a cuboid is a commonly tested formula that many aspirants forget.

• Cube: all edges equal, 6 equal square faces
• Cuboid: 3 pairs of rectangular faces
• Lateral Surface Area of cuboid = 2h(l+b), useful when top/bottom are open
• Volume of cube = a^3; very common in questions about ratio of volumes after side changes
• Diagonal of cube = a*sqrt(3); diagonal of cuboid = sqrt(l^2 + b^2 + h^2)
• If side of cube is doubled, volume becomes 8 times — this ratio trick is directly asked

Cylinder

A cylinder has two circular faces and one curved surface. Think of a can of paint. SSC CGL loves questions where water flows through a pipe (cylinder) to fill a tank, or a roller (cylinder) paints a road — that uses Curved Surface Area. Always check if the question asks for open or closed cylinder.

• Curved Surface Area (CSA) = 2*pi*r*h — used for pipes, rollers, road painting
• Total Surface Area = 2*pi*r*(r+h) — used when both circular ends are included
• Volume = pi*r^2*h
• If height doubles and radius halves, volume becomes pi*(r/2)^2*(2h) = pi*r^2*h/2 — volume is halved
• Hollow cylinder volume = pi*h*(R^2 - r^2) where R=outer radius, r=inner radius

Cone

A cone has a circular base and a pointed apex. The slant height l connects the apex to the edge of the base. Remember: slant height is NOT the same as vertical height. Use Pythagoras: l^2 = r^2 + h^2. Most mistakes happen when a question gives vertical height but you use it directly as slant height in CSA formula.

• Slant height l = sqrt(r^2 + h^2) — always calculate this first if not given
• CSA = pi*r*l — think of it as unrolling the cone into a sector
• TSA = pi*r*(r + l) — includes circular base
• Volume = (1/3)*pi*r^2*h — one-third of cylinder with same base and height
• Cone + hemisphere combo questions appear frequently in Tier 2

Sphere and Hemisphere

A sphere has no flat face. A hemisphere is exactly half a sphere but has one flat circular face at the bottom. This flat face is often forgotten when computing Total Surface Area of hemisphere. Think of a bowl: the inside curve plus the flat circular rim at the top.

• Sphere TSA = 4*pi*r^2 — there is no separate CSA vs TSA for sphere
• Hemisphere CSA = 2*pi*r^2 (curved part only)
• Hemisphere TSA = 3*pi*r^2 (curved + flat circular base)
• Volume of sphere = (4/3)*pi*r^3
• Volume of hemisphere = (2/3)*pi*r^3
• If radius is doubled, surface area becomes 4 times and volume becomes 8 times

Melting and Recasting (Conversion Problems)

These are the most common SSC CGL 3D problems. One solid is melted and recast into another. The key principle: Volume remains constant (assuming no wastage). Set Volume1 = Volume2 and solve for the unknown dimension. Wastage, if mentioned, means final volume = original volume x (1 - wastage%).

• Core rule: Volume of original = Volume of new shape (no wastage)
• With wastage: Volume of new = Volume of original x (1 - wastage/100)
• Number of small shapes = Volume of large / Volume of one small shape
• Always convert units to the same before computing
• Common combo: cylinder melted into cones, sphere melted into small spheres, cone into hemisphere

• Using vertical height h instead of slant height l in the cone CSA formula — always calculate l = sqrt(r^2+h^2) first
• Forgetting the flat circular base of hemisphere in TSA — hemisphere TSA is 3*pi*r^2, not 2*pi*r^2
• Confusing Lateral/Curved Surface Area with Total Surface Area when the question asks for cost of painting only the sides
• Not converting units — mixing cm and m in the same problem leads to wrong volume by a factor of 10^6
• In cube painting and cutting problems, miscounting edge cubes — corners have 3 painted faces, edges have 2, faces have 1, and inner cubes have 0

• Cone volume = one-third of cylinder: CONE is a CUT cylinder — cut away two-thirds, keep one-third
• Hemisphere TSA = 3*pi*r^2: think 3 as Three-part bowl: 2 curved + 1 flat = 3
• Slant height mnemonic: SLOPE needs Pythagoras — l^2 = r^2 + h^2, just like a right triangle on the side of the cone
• For recasting: VOLUME IS IMMORTAL — it never changes when you melt and recast (unless wastage is given)
• Sphere surface area = 4 x circle area: one sphere wraps around exactly 4 circles of the same radius

• SSC CGL Tier 1 typically has 2-3 direct formula-application questions on cylinder and cone. Memorize formulas cold — no time to derive.
• Tier 2 Paper 1 (Math) often has a combined solid question like a cone placed on a hemisphere or cylinder — practice finding total surface area by adding CSA of each part (do not double-count shared base).
• Recasting problems (melting one shape into another) appear almost every year. These are 1-2 minute questions if you remember Volume1 = Volume2 — do not overthink.
• Cube painting and cutting is a pure logic question — no formula needed, just visualize layers. Corner=3 faces, Edge=2 faces, Face=1 face, Inside=0 faces. Draw it once and remember for life.
• Use pi = 22/7 when radius is a multiple of 7, and pi = 3.14 otherwise — SSC questions are designed for clean answers with the right pi value.