Surface Areas and Volumes · Lesson 4 of 4
Chapter Summary and Practice
“A silo is just a cylinder wearing a conical hat, but missing its height in the formula will cost you half your exam points.”
• By the end of this lesson, you should be able to. • Revise the entire chapter quickly and systematically. • Choose the correct surface-area or volume strategy. • Solve mixed practice questions confidently. • Check whether a face should be counted, removed or ignored.
This chapter is all about breaking a complicated solid into simpler parts. Once you identify the basic solids and decide whether the problem is about outer covering or inner space, the rest becomes a careful use of formulas.
Introduction Snapshot
Combination solids are formed by joining basic solids such as cones, cylinders, cuboids and hemispheres, or by cutting one solid out of another. The first step in every question is to identify those parts clearly.
Surface Area Summary
For surface area, count only the surfaces that are exposed. Hidden faces where two solids join are not counted. If a cavity is cut, its inside curved surface becomes part of the remaining outer surface.
| Situation | How to think |
|---|---|
| Two solids joined together | Add only exposed curved/flat parts; ignore hidden common face |
| One solid fixed on another | Subtract the covered patch, then add exposed curved area |
| Cavity cut out | Count the new exposed inner curved surface |
| Painting/polishing/covering | This is a surface area question |
Volume Summary
For volume, hidden faces do not matter. Add the volumes of solids that are present and subtract the volumes of holes, cavities, depressions or bulges that are removed from the capacity.
| Situation | How to think |
|---|---|
| Solid made by joining parts | Add the volumes of the parts |
| Cavity or depression | Subtract the removed volume |
| Container with raised bottom | Actual capacity = apparent capacity − raised portion |
| Mass problem | Mass = density × volume |
Formula Recap
| Solid | Useful formulas |
|---|---|
| Cylinder | CSA = 2πrh, TSA = 2πr(h + r), V = πr²h |
| Cone | CSA = πrl, TSA = πr(l + r), V = ⅓πr²h |
| Sphere | SA = 4πr², V = ⁴⁄₃πr³ |
| Hemisphere | CSA = 2πr², TSA = 3πr², V = ²⁄₃πr³ |
| Cuboid | TSA = 2(lb + bh + hl), V = lbh |
Guided Solved Problems
Problem
A toy is made of a cone on a hemisphere. Both have radius 2.5 cm, and the cone has slant height 6 cm. Find the area to be painted.
- 1.Area to be painted = CSA of cone + CSA of hemisphere.
- 2.CSA of cone = πrl = π × 2.5 × 6 = 15π cm².
- 3.CSA of hemisphere = 2πr² = 2π × (2.5)² = 12.5π cm².
- 4.Total area = 15π + 12.5π = 27.5π cm².
Problem
A cylindrical glass has radius 2 cm and height 9 cm. A hemispherical bulge of the same radius is present at the bottom. Find the actual capacity.
- 1.Apparent capacity = πr²h = π × 2² × 9 = 36π cm³.
- 2.Volume of hemispherical bulge = ²⁄₃πr³ = ²⁄₃π × 2³ = 16π/3 cm³.
- 3.Actual capacity = 36π − 16π/3 = 92π/3 cm³.
Problem
A solid cylinder of radius 4 cm and height 10 cm has a conical cavity of the same radius and height cut from the top. Find the volume of the remaining solid.
- 1.Volume of cylinder = πr²h = π × 4² × 10 = 160π cm³.
- 2.Volume of cone removed = ⅓πr²h = ⅓π × 4² × 10 = 160π/3 cm³.
- 3.Remaining volume = 160π − 160π/3 = 320π/3 cm³.
Before solving any chapter question, ask: • Which basic solids are present? • Is the question about surface area or volume? • Which faces are exposed? • Is any part hidden or removed? • Are all dimensions in the same unit?
Quiz
When two solids are joined, which area is usually NOT counted in the outer surface area?
A cavity cut out from a solid affects the volume by:
A capsule is best modelled as:
If a hemisphere is fixed on the top face of a cube, the exposed area includes:
A raised bulge at the bottom of a container makes the actual capacity:
Practice Problems
- Two cubes each of edge 5 cm are joined face to face. Find the total outer surface area of the new solid.
- A vessel consists of a hollow hemisphere topped by a hollow cylinder. The hemisphere radius is 6 cm and the total inner height is 16 cm. Find the inner surface area.
- A toy is in the form of a cone of radius 3 cm mounted on a hemisphere of the same radius. If the total height is 12 cm, find its outer surface area.
- A capsule is in the shape of a cylinder with two hemispheres at its ends. Its total length is 18 mm and its diameter is 6 mm. Find its outer surface area.
- A tent is in the form of a cylinder with a conical top. The cylindrical part has height 2 m and diameter 3 m, and the conical top has slant height 2.5 m. Find the canvas area needed if the base is open.
- A solid consists of a cone standing on a hemisphere, both having radius 3 cm. The cone height is 3 cm. Find the volume of the solid.
- A model consists of a cylinder with two cones attached at its ends. The diameter is 4 cm, cylinder length is 8 cm and each cone height is 2 cm. Find the total volume.
- A wooden pen stand is a cuboid 15 cm × 10 cm × 3.5 cm with four conical depressions, each of radius 0.5 cm and depth 1.4 cm. Find the volume of wood left.
- A cylindrical glass of radius 2.5 cm and height 10 cm has a hemispherical bulge at the base. Find its apparent and actual capacities.
- A solid iron pole is formed by two cylinders. Find its mass if all dimensions are given and 1 cm³ of iron weighs 8 g.
Key Takeaways
• Combination solids become easy when you split them into familiar parts. • Surface area means exposed surfaces only. • Volume means add the parts present and subtract the parts removed. • Diagrams, formulas and units must work together. • Careful structure beats memorising random tricks.
Previous · Lesson 3
Volume of a Combination of Solids
Next
End of chapter