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Class 10 Surface Areas and Volumes: cheat sheet
CBSE Mathematics. Solids made by joining cylinders, cones, hemispheres and cubes: add the volumes, and add only the surfaces that stay visible.
Key ideas
- The volume of a combined solid is the sum of the volumes of its parts.
- The surface area of a combined solid is the sum of the surfaces you can still see; the faces where parts are joined are hidden.
- A hemisphere on a cylinder hides the top of the cylinder and the flat face of the hemisphere.
- When a part is scooped out, subtract its volume; for the area, take away the face that was cut and add the inside surface.
- Slant height of a cone: l = √(r² + h²).
- Convert units at the end: 1000 cm³ = 1 litre, 1 m³ = 1000 litres.
Remember
| Cylinder | curved area 2πrh; volume πr²h |
| Cone | curved area πrl; volume (1/3)πr²h |
| Hemisphere | curved area 2πr²; volume (2/3)πr³ |
| Sphere | area 4πr²; volume (4/3)πr³ |
Watch out
- Adding the total surface areas of the parts, so the joined faces are counted.
- Using the height of a cone where its slant height is needed (or the other way round).
- Mixing cm and m in one calculation.
Worked example
A hemispherical hollow of radius 7 cm is scooped out of the top of a solid wooden block 33 cm × 32 cm × 15 cm. Find the volume of wood left (π = 22/7).
Working: 15,840 − (2/3) × 22/7 × 7³ = 15,840 − 718 2/3
Answer: 15,121 1/3 cm³
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