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 |