Mensuration of Solids

Overview

Mensuration of solids deals with calculating the surface area and volume of three-dimensional figures. This topic carries significant weightage in MP TET Varg-2, typically contributing 3–5 questions in the mathematics section. Questions test both formula recall and application to real-world scenarios like water tanks, containers, and construction materials.

Mastery requires memorising formulas for five standard solids—cube, cuboid, cylinder, cone, and sphere—and understanding when to use total surface area versus curved/lateral surface area. Many questions involve combined figures or conversions between units, so dimensional analysis skills are equally important. This topic also connects to EVS themes like water conservation (tank capacity) and practical mathematics in daily life.

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Key Concepts

• **Surface Area** measures the total outer covering of a solid. It determines how much material is needed to wrap, paint, or cover the object.

• **Curved Surface Area (CSA)** or **Lateral Surface Area (LSA)** excludes the top and bottom faces—relevant when calculating material for open containers or cylindrical pipes.

• **Total Surface Area (TSA)** includes all faces—top, bottom, and curved/lateral surfaces combined.

• **Volume** measures the space occupied by a solid or the capacity it can hold. Volume determines how much liquid, grain, or material fits inside.

• **Units matter**: Surface area is in square units (cm², m²); volume is in cubic units (cm³, m³). 1 litre = 1000 cm³ = 0.001 m³.

• **Hemisphere** is half a sphere. Its formulas are derived by halving sphere values and adjusting for the circular base.

• **Slant height (l)** is crucial for cones—the distance from the apex to any point on the circular edge, related to radius and height by the Pythagorean theorem.

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Formulas / Key Facts

### Cube (side = a) | Measure | Formula | |---------|---------| | Volume | a³ | | TSA | 6a² | | LSA | 4a² | | Diagonal | a√3 |

### Cuboid (length = l, breadth = b, height = h) | Measure | Formula | |---------|---------| | Volume | l × b × h | | TSA | 2(lb + bh + hl) | | LSA | 2h(l + b) | | Diagonal | √(l² + b² + h²) |

### Cylinder (radius = r, height = h) | Measure | Formula | |---------|---------| | Volume | πr²h | | CSA | 2πrh | | TSA | 2πr(r + h) |

### Cone (radius = r, height = h, slant height = l) | Measure | Formula | |---------|---------| | Slant height | l = √(r² + h²) | | Volume | (1/3)πr²h | | CSA | πrl | | TSA | πr(r + l) |

### Sphere (radius = r) | Measure | Formula | |---------|---------| | Volume | (4/3)πr³ | | Surface Area | 4πr² |

### Hemisphere (radius = r) | Measure | Formula | |---------|---------| | Volume | (2/3)πr³ | | CSA | 2πr² | | TSA | 3πr² |

**Quick conversions**: 1 m³ = 1000 litres; 1 litre = 1000 cm³

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Worked Examples

### Example 1: Cube Problem **A cube has a total surface area of 294 cm². Find its volume.**

Step 1: TSA of cube = 6a² 6a² = 294 a² = 49 a = 7 cm

Step 2: Volume = a³ = 7³ = **343 cm³**

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### Example 2: Cylinder Capacity **A cylindrical tank has radius 7 m and height 10 m. How many litres of water can it hold? (Use π = 22/7)**

Step 1: Volume = πr²h = (22/7) × 7² × 10 = (22/7) × 49 × 10 = 22 × 70 = 1540 m³

Step 2: Convert to litres 1 m³ = 1000 litres Volume = 1540 × 1000 = **15,40,000 litres**

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### Example 3: Cone with Slant Height **A cone has radius 6 cm and height 8 cm. Find its curved surface area and volume. (Use π = 3.14)**

Step 1: Find slant height l = √(r² + h²) = √(36 + 64) = √100 = 10 cm

Step 2: CSA = πrl = 3.14 × 6 × 10 = **188.4 cm²**

Step 3: Volume = (1/3)πr²h = (1/3) × 3.14 × 36 × 8 = (1/3) × 904.32 = **301.44 cm³**

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### Example 4: Sphere to Hemisphere Comparison **A solid sphere of radius 3 cm is melted and recast into a hemisphere. Find the radius of the hemisphere.**

Step 1: Volume of sphere = (4/3)πr³ = (4/3)π × 27 = 36π cm³

Step 2: Let hemisphere radius = R Volume of hemisphere = (2/3)πR³

Step 3: Equate volumes (material conserved) (2/3)πR³ = 36π R³ = 54 R = ³√54 ≈ **3.78 cm**

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Common Mistakes

• **Confusing CSA and TSA** → CSA excludes circular faces; TSA includes everything. Read whether the container is "open" or "closed."

• **Forgetting to calculate slant height for cone** → Students directly use height in CSA formula. Always compute l = √(r² + h²) first.

• **Unit conversion errors** → Mixing cm and m in the same problem leads to answers off by factors of 1000 or 1,000,000. Convert all measurements to the same unit before calculating.

• **Using diameter instead of radius** → Many word problems give diameter. Divide by 2 immediately to get radius before substituting.

• **Volume of cone = πr²h** → Wrong! Cone volume is one-third of cylinder volume. Always include the (1/3) factor.

• **Hemisphere TSA = 2πr²** → Wrong! The flat circular base must be added. TSA = 2πr² + πr² = 3πr².

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Quick Reference

• **Cube volume**: a³ | **Cube TSA**: 6a²

• **Cuboid volume**: lbh | **Cuboid TSA**: 2(lb + bh + hl)

• **Cylinder volume**: πr²h | **Cylinder TSA**: 2πr(r + h)

• **Cone volume**: (1/3)πr²h | **Cone CSA**: πrl (where l = √(r² + h²))

• **Sphere volume**: (4/3)πr³ | **Sphere SA**: 4πr²

• **1 m³ = 1000 litres** — memorise for capacity problems