3-D Mensuration — Study Notes

Overview

3-D Mensuration forms a core component of SSC CHSL Tier 1 Quantitative Aptitude, consistently contributing 2–4 questions per exam. This topic tests your ability to calculate surface areas and volumes of three-dimensional objects—cubes, cuboids, cylinders, cones, and spheres. Unlike 2-D mensuration which deals with flat shapes, here you work with solid objects that occupy space.

Mastery of this topic requires two things: memorizing the correct formulas (many students confuse curved surface area with total surface area) and applying them accurately under time pressure. The good news is that questions follow predictable patterns—direct formula application, converting units, or combining two shapes. A solid grasp of 3-D mensuration also helps in Data Interpretation questions involving container capacities and real-world problem-solving scenarios.

Expect straightforward calculations mixed with word problems involving melting/recasting solids, filling tanks, or comparing volumes. Speed and accuracy in applying the right formula to the right shape determine your score here.

Key Concepts

• **Surface Area vs Volume**: Surface area measures the total outer area of a solid (units: cm², m²). Volume measures the space occupied inside (units: cm³, m³, litres). Don't confuse them—surface area for painting/wrapping problems, volume for capacity/filling problems.

• **Curved vs Total Surface Area**: For cylinders and cones, curved surface area (CSA) covers only the rounded part, while total surface area (TSA) includes the base(s). Cubes and cuboids have only total surface area since all faces are flat.

• **Dimensional Relationships**: Cube is a special cuboid where length = breadth = height. Cylinder and cone share circular bases but differ in shape—cylinder has uniform cross-section, cone tapers to a point.

• **Slant Height in Cones**: The slant height (l) is the distance from apex to the edge of the base. It forms a right triangle with height (h) and radius (r): l² = h² + r². Many students forget to use slant height for cone surface area.

• **Unit Conversions**: 1 m³ = 1000 litres. 1 cm³ = 1 ml. Always convert all measurements to the same unit before calculating. Mixed units are a common exam trap.

• **Conservation of Volume**: When one solid is melted and recast into another, volumes remain equal but surface areas change. This principle solves recasting problems efficiently.

Formulas / Key Facts

**Cube (edge = a)**

• Total Surface Area = 6a²
• Volume = a³
• Diagonal of cube = a√3

**Cuboid (length = l, breadth = b, height = h)**

• Total Surface Area = 2(lb + bh + hl)
• Volume = l × b × h
• Diagonal of cuboid = √(l² + b² + h²)

**Cylinder (radius = r, height = h)**

• Curved Surface Area = 2πrh
• Total Surface Area = 2πrh + 2πr² = 2πr(h + r)
• Volume = πr²h

**Cone (radius = r, height = h, slant height = l)**

• Curved Surface Area = πrl (use slant height)
• Total Surface Area = πrl + πr² = πr(l + r)
• Volume = (1/3)πr²h (use perpendicular height)
• Slant height: l = √(h² + r²)

**Sphere (radius = r)**

• Surface Area = 4πr²
• Volume = (4/3)πr³

**Hemisphere (radius = r)**

• Curved Surface Area = 2πr²
• Total Surface Area = 2πr² + πr² = 3πr²
• Volume = (2/3)πr³

Take π = 22/7 or 3.14 as specified in the question.

Worked Examples

**Example 1**: A cuboid water tank is 5 m long, 4 m wide and 3 m deep. How many litres of water can it hold when full?

*Solution*:

• Volume = l × b × h = 5 × 4 × 3 = 60 m³
• Convert to litres: 60 m³ = 60 × 1000 = 60,000 litres
• Answer: 60,000 litres

**Example 2**: Find the total surface area of a cylinder with radius 7 cm and height 10 cm. (Use π = 22/7)

*Solution*:

• TSA = 2πr(h + r)
• TSA = 2 × (22/7) × 7 × (10 + 7)
• TSA = 2 × 22 × 17 = 44 × 17 = 748 cm²
• Answer: 748 cm²

**Example 3**: A cone has base radius 6 cm and height 8 cm. Find its curved surface area.

*Solution*:

• First find slant height: l = √(h² + r²) = √(8² + 6²) = √(64 + 36) = √100 = 10 cm
• CSA = πrl = (22/7) × 6 × 10 = 1320/7 ≈ 188.57 cm²
• Answer: 188.57 cm² (or leave as 1320/7 if exact value needed)

**Example 4**: A metallic sphere of radius 3 cm is melted and recast into a cylinder of radius 2 cm. Find the height of the cylinder.

*Solution*:

• Volume of sphere = Volume of cylinder
• (4/3)πr³ = πR²H (where r = 3 for sphere, R = 2 for cylinder)
• (4/3) × 3³ = 2² × H
• (4/3) × 27 = 4H
• 36 = 4H
• H = 9 cm
• Answer: Height = 9 cm

Common Mistakes

**Using height instead of slant height for cone CSA** → The curved surface area of a cone uses slant height (l), not perpendicular height (h). Always calculate l = √(h² + r²) first if not given directly.

**Forgetting to include base(s) in total surface area** → Students often calculate only curved surface area for cylinders/cones. Remember: TSA includes all surfaces. For a cylinder, add both circular bases (2πr²). For a cone, add one base (πr²).

**Mixing up cube and cuboid formulas** → A cube is a special cuboid but has simpler formulas. For a cube with edge 'a', volume is a³ not a × a × a written separately. Surface area is 6a², not 2(a² + a² + a²).

**Incorrect unit conversions** → 1 m = 100 cm, but 1 m³ = 1,000,000 cm³ (not 100). For volume to litres: 1 m³ = 1000 litres, 1 cm³ = 1 ml. Convert all dimensions to the same unit before calculating.

**Using volume formula when surface area is asked (and vice versa)** → Read the question carefully. "Paint the outer surface" needs surface area. "How much water can it hold" needs volume. These are completely different calculations with different units.

Quick Reference

• **Cube**: All edges equal → Volume = a³, Surface = 6a²
• **Cylinder volume** = πr²h; add 2πr² to curved surface for total surface
• **Cone needs slant height** for curved surface: CSA = πrl where l = √(h² + r²)
• **Sphere surface** = 4πr²; Volume = (4/3)πr³ (the "four-thirds" formula)
• **1 m³ = 1000 litres**; always unify units before calculating
• **Melting problems**: Set volumes equal, surface areas change