Surface Areas and Volumes
About 6 min · Mathematical Reasoning · English
Time worked out from the lesson's 426 words at 150 words a minute, plus a minute for each question.
Surface Areas and Volumes is part of the SOF IMO Mathematical Reasoning section, directly testing your ability to visualize three-dimensional objects and apply precise formulas. This topic can appear in multi-step word problems requiring you to calculate material costs, capacities, or conversions between different solid shapes.
Key ideas
- Lateral vs. Total Surface Area: Lateral surface area (LSA or CSA—curved surface area) covers only the sides, excluding top and bottom bases. Total surface area (TSA) includes all exposed faces. For a cylinder, CSA = 2πrh; TSA = 2πr(r + h).
- Volume as Capacity: Volume measures the space inside a solid, directly translating to capacity for containers. 1 cubic metre = 1000 litres, and 1 cubic centimetre = 1 millilitre—critical for word problems involving water tanks or paint coverage.
- Composite Solids: Real IMO problems present combinations like a tent (cone + cylinder), a capsule (cylinder + two hemispheres), or a toy (cone on hemisphere). Calculate each part separately, then sum for total volume; for surface area, exclude contact regions.
- Dimension Relationships: Radius (r), diameter (d = 2r), height (h), and slant height (l) are interconnected. For a cone, l² = r² + h² by Pythagoras. Always check which measurements the problem provides versus what the formula needs.
- Unit Conversions: Problems may mix units—radius in cm, height in m, asking answer in litres. Convert everything to one unit system before plugging into formulas. 1 m = 100 cm; 1 m³ = 10⁶ cm³.
Quick reference
- Cube: Volume = a³, TSA = 6a². If edge doubles, volume increases 8×, surface area 4×.
- Cylinder: CSA = 2πrh, TSA = 2πr(r + h). Open top/bottom? Subtract πr² per missing base.
- Cone slant height: l = √(r² + h²). Must calculate before finding TSA.
- Sphere: SA = 4πr², Volume = (4/3)πr³. Hemisphere volume is exactly half, but TSA = 3πr².
- Melting problems: Volume before = Volume after. Solve for unknown dimension.
Check yourself: 3 questions
1. A rectangular water tank has dimensions 5 m × 3 m × 2 m. If the tank is completely filled with water, what is the volume of water in the tank (in cubic metres)?
2. A cylindrical pillar has a diameter of 28 cm and height of 5 m. Find the curved surface area of the pillar. (Use π = 22/7)
3. A solid metallic hemisphere of radius 7 cm is melted and recast into a solid cone of base radius 7 cm. What is the height of the cone? (Use π = 22/7)
This lesson is cut from the topic's note, not rewritten. Fact-checked on 21 Oct 2026: an AI-assisted check of this note against cited sources; the errors it found were corrected that day. Some errors may remain. The questions are from Shishya's validated question bank.
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