Surface Areas and Volumes
CBSE Class 9 · Mathematics · Notes, formulas and practice questions
Surface areas and volumes of solid figures for CBSE Class 9: cuboid, cube, cylinder, cone, sphere and hemisphere. Learn the formulas, how to choose between curved and total area, and worked numericals for capacity and cost problems.
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This chapter extends the ideas of area and perimeter of plane figures to three-dimensional solids. For every solid you must learn two separate measures: surface area, the total region of the outer faces that can be painted or covered, and volume, the amount of space inside that holds water, grain or air. These two measures usually need different formulas, and the first skill is to tell which one a question demands.
The cube and cuboid come straight from rectangular sheets. A cuboid has six rectangular faces, so its total surface area is the sum of the three pairs of opposite faces, 2(lb + bh + hl); its lateral surface area leaves out the top and bottom. A cube is the special case where every edge is equal. The cylinder can be imagined as a rectangle rolled into a tube, which explains why its curved surface area equals the rectangle's area, 2πrh.
The cone is a pyramid-like solid with a circular base. The key extra idea is the slant height, the distance along the sloping surface. Since the radius, vertical height and slant height form a right triangle, l² = r² + h². The sphere needs no such triangle, but the hemisphere is simply half a sphere with a flat circular lid, and that lid must be added back when you find total surface area.
In exercises you are often asked to find a cost, a capacity or the area to be painted. Such questions tie the formula to daily life: multiply the area by the rate, or change cubic metres into litres using 1 m³ = 1000 litres. Look carefully at units. A very common slip is to mix centimetres and metres, or to forget that a closed cylinder has two circular ends whereas an open container has only one.
Key terms
- Surface area
- The total area of the outside faces of a solid, measured in square units such as cm² or m². In problems it tells you the amount of material, paint or paper needed to cover the solid.
- Curved or lateral surface area
- Area of only the curved or side surfaces, not the flat top and bottom. For example, the curved surface area of a cylinder excludes its two circular ends, and the lateral surface area of a cuboid excludes its top and bottom.
- Total surface area
- The sum of the areas of all exterior surfaces, curved as well as flat. For a solid with a base, like a cone or hemisphere, the base is included.
- Volume
- The space occupied by a solid, measured in cubic units. For containers, the volume tells the capacity: 1 m³ = 1000 litres and 1 cm³ = 1 mL.
- Right circular cylinder
- A cylinder whose flat ends are circles and whose straight line joining the centres of the two circles is perpendicular to those circles. Its curved surface, if unrolled, becomes a rectangle of length 2πr and height h.
- Right circular cone
- A solid that stands upright on a circular base with its vertex directly above the centre of the base. Its curved surface tapers uniformly to the vertex, and its radius, vertical height and slant height obey l² = r² + h².
- Slant height
- The shortest distance along the curved surface of a cone from the vertex to a point on the rim of the base. It is measured on the slant, not vertically; in a right cone it equals √(r² + h²).
- Sphere and hemisphere
- A sphere is a perfectly round solid with every point on its surface equally far from its centre. A hemisphere is exactly half of a sphere cut through its centre; it has one curved surface and one flat circular base.
Formula sheet
| What | Formula | Notes |
|---|---|---|
| Volume of a cube | V = a³ | a is the length of an edge of the cube. |
| Total surface area of a cube | A = 6a² | a is the length of an edge. |
| Lateral surface area of a cube | A = 4a² | a is the length of an edge. |
| Volume of a cuboid | V = lbh | l is length, b is breadth, h is height. |
| Total surface area of a cuboid | A = 2(lb + bh + hl) | l is length, b is breadth, h is height. |
| Lateral surface area of a cuboid | A = 2h(l + b) | h is height, l is length, b is breadth. |
| Curved surface area of a right circular cylinder | A = 2πrh | r is the radius of the base and h is the height. |
| Total surface area of a right circular cylinder | A = 2πr(r + h) | r is the base radius and h is the height; both circular ends are included. |
| Volume of a right circular cylinder | V = πr²h | r is the base radius and h is the height. |
| Slant height of a cone | l = √(r² + h²) | r is the base radius, h is the vertical height and l is the slant height. |
| Curved surface area of a cone | A = πrl | r is the base radius and l is the slant height. |
| Total surface area of a cone | A = πr(l + r) | r is the base radius and l is the slant height; the circular base is included. |
| Volume of a cone | V = (1/3)πr²h | r is the base radius and h is the vertical height. |
| Surface area of a sphere | A = 4πr² | r is the radius of the sphere. |
| Volume of a sphere | V = (4/3)πr³ | r is the radius of the sphere. |
| Curved surface area of a hemisphere | A = 2πr² | r is the radius; this excludes the flat circular base. |
| Total surface area of a hemisphere | A = 3πr² | r is the radius; the flat circular base is included. |
| Volume of a hemisphere | V = (2/3)πr³ | r is the radius of the hemisphere. |
Practice questions with answers
1. Explain the difference between the lateral surface area and the total surface area of a cuboid.
The lateral surface area covers only the four side faces, so for a cuboid of length l, breadth b and height h it is 2h(l + b). The total surface area includes the two horizontal faces as well, so all six faces together give 2(lb + bh + hl). In a room, the lateral area is the area of the four walls, while the total area adds the ceiling and floor.
2. Find the total surface area and volume of a cube whose edge is 9 cm.
Total surface area = 6a² = 6 × 9² = 6 × 81 = 486 cm². Volume = a³ = 9³ = 729 cm³.
3. A cuboidal tank is 4 m long, 2.5 m wide and 1.5 m deep. How many litres of water can it hold?
Volume = lbh = 4 × 2.5 × 1.5 = 15 m³. Since 1 m³ = 1000 litres, the capacity = 15 × 1000 = 15,000 L.
4. The curved surface area of a right circular cylinder is 440 cm² and its height is 20 cm. Find the radius of its base. (Take π = 22/7)
For a cylinder, curved surface area = 2πrh. Substituting, 440 = 2 × (22/7) × r × 20 = (880/7)r. Solving gives r = 440 × 7/880 = 3.5 cm.
5. A cylindrical tin has a base radius of 7 cm and a height of 12 cm. Find its volume. (Take π = 22/7)
Volume = πr²h = (22/7) × 7² × 12 = (22/7) × 49 × 12 = 22 × 7 × 12 = 1848 cm³.
6. A cone has a base radius of 10 cm and a slant height of 28 cm. Find its curved surface area. (Take π = 22/7)
Curved surface area = πrl = (22/7) × 10 × 28 = (22/7) × 280 = 22 × 40 = 880 cm².
7. The base radius of a cone is 7 cm and its height is 24 cm. Find its slant height and volume. (Take π = 22/7)
Slant height l = √(r² + h²) = √(7² + 24²) = √(49 + 576) = √625 = 25 cm. Volume = (1/3)πr²h = (1/3) × (22/7) × 49 × 24 = (22 × 7 × 24)/3 = 1232 cm³.
8. Find the surface area of a sphere whose diameter is 14 cm. (Take π = 22/7)
Radius = 14/2 = 7 cm. Surface area = 4πr² = 4 × (22/7) × 7² = 4 × (22/7) × 49 = 4 × 154 = 616 cm².
9. A solid hemisphere has radius 3 cm. Find its total surface area. (Take π = 3.14)
Total surface area = 3πr² = 3 × 3.14 × 3² = 3 × 3.14 × 9 = 3 × 28.26 = 84.78 cm². This includes the curved surface (2πr²) and the flat circular base (πr²).
10. The curved surface of a cylindrical pillar of radius 42 cm and height 4 m is to be painted. If the rate is ₹40 per square metre, find the cost. (Take π = 22/7)
Convert radius to metres: 42 cm = 0.42 m. Curved surface area = 2πrh = 2 × (22/7) × 0.42 × 4 = 10.56 m². Cost = 10.56 × 40 = ₹422.40.
Studying Surface Areas and Volumes?
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