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.
### 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.**
**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.
A cube has an edge of 10 cm. What is the total surface area of the cube?
Q2 · Mensuration of Solids · MEDIUM
A cylindrical water tank has radius 7 m and height 10 m. What is the volume of water it can hold? (Take pi = 22/7)
Q3 · Mensuration of Solids · MEDIUM
A cone has base radius 3 cm and slant height 5 cm. What is the curved surface area of the cone? (Take pi = 22/7)
Q4 · Mensuration of Solids · HARD
A solid metal sphere of radius 6 cm is melted and recast into a cylinder of radius 3 cm. What is the height of the cylinder? (Take pi = 22/7)
Q5 · Mensuration of Solids · MEDIUM
A cylindrical water tank has a radius of 3.5 m and height of 6 m. If it is filled completely with water, how many litres of water does it hold? (Use π = 22/7)