CBSE Class 10 Mathematics • Chapter 12 • Revision 2026
Surface Areas and Volumes Class 10
Revise mensuration with our Surface Areas and Volumes Class 10 notes and mind map PDF. The PDF contains notes and a mind map for organising your revision. Use this page to review important formulas, understand combined solids and practise worked examples with clear steps.
How much material is needed to make a tent? How much liquid can a container hold? These questions involve different measurements. Surface area measures a solid’s surfaces, while volume measures the space it occupies. In Class 10 Maths Surface Areas and Volumes, the main challenge is recognising the basic shapes inside an object and deciding which surfaces or volumes belong in the calculation.
These revision notes explain cylinders, cones, spheres, hemispheres, cubes and cuboids before applying their formulas to combinations. Read the explanations, practise the examples and use the mind map to recall the method before attempting your textbook questions.
Surface Areas and Volumes Class 10 Notes: Basic Concepts
- Curved surface area (CSA): The area of a solid’s curved surface, excluding any flat circular bases.
- Lateral surface area (LSA): The area of the side surfaces, excluding the chosen top and bottom bases.
- Total surface area (TSA): The area of all surfaces forming the solid’s boundary.
- Volume: The amount of three-dimensional space occupied by a solid.
- Capacity: The amount a container can hold, calculated using its internal dimensions.
- Slant height: The distance along a cone’s sloping surface from its vertex to the edge of its base.
Choose the Measurement Before the Formula
Painting, polishing, wrapping and covering usually involve surface area. Filling a tank, measuring solid material or finding a container’s capacity involves volume. Read which surfaces are included: an open container, a closed container and a tent without a floor need different area calculations.
Surface Areas and Volumes Class 10 All Formulas
Here, r means radius, h means perpendicular height, s means a cone’s slant height and a means the edge of a cube. For a cuboid, L, B and H represent length, breadth and height.
| Solid | Curved or Lateral Surface Area | Total Surface Area | Volume |
|---|---|---|---|
| Cube | 4a2 | 6a2 | a3 |
| Cuboid | 2H(L + B) | 2(LB + BH + HL) | LBH |
| Right circular cylinder | 2πrh | 2πr(h + r) | πr2h |
| Right circular cone | πrs | πr(s + r) | (1/3)πr2h |
| Sphere | 4πr2 | 4πr2 | (4/3)πr3 |
| Hemisphere | 2πr2 | 3πr2 | (2/3)πr3 |
For a right circular cone, the radius, perpendicular height and slant height form a right-angled triangle: s2 = r2 + h2. Many textbooks use the letter l for slant height. The relationship remains the same.
Remember the Units
Surface area is measured in square units such as cm2 or m2. Volume is measured in cubic units such as cm3 or m3. 1 cm3 = 1 mL, 1000 cm3 = 1 litre and 1 m3 = 1000 litres.
Surface Area of a Combination of Solids
When solids are joined, their contact surfaces become hidden inside the combined object. Calculate the exposed boundary rather than adding every original surface. For example, when a cone is joined to a hemisphere along equal circular bases, those bases are internal. The outside consists of the cone’s curved surface and the hemisphere’s curved surface.
A useful general method is to add the original total surface areas and subtract both copies of each joined contact area. Alternatively, list the exposed surfaces directly. This also helps when a smaller solid is attached to a larger base and part of that base remains visible.
Example 1: A Cone Joined to a Hemisphere
A solid toy consists of a cone joined to a hemisphere along their equal bases. The common radius is 7 cm and the cone’s height is 24 cm. Find the exposed surface area. Use π = 22/7.
Slant height of cone:
s2 = 72 + 242
s2 = 49 + 576
s2 = 625
s = 25 cm
Exposed area = cone CSA + hemisphere CSA
= πrs + 2πr2
= (22/7) × 7 × 25 + 2 × (22/7) × 49
= 550 + 308
= 858 cm2
The common circular base is inside the toy, so it is excluded.
Volume of a Combination of Solids
Add the volumes of the component solids when they occupy separate regions and meet only at their boundaries. If a cavity is removed, subtract its volume from the original solid. Identify the dimensions of each component before substituting values: the total length of a combined object may differ from its cylinder’s length or its cone’s height.
Example 2: A Cylinder with Two Hemispherical Ends
A capsule-shaped solid has total length 20 cm and radius 3 cm. Find its volume in terms of π.
Each hemispherical end contributes 3 cm to the total length.
Cylinder length = 20 − 2 × 3
= 14 cm
Combined volume = cylinder volume + sphere volume
= πr2h + (4/3)πr3
= π × 9 × 14 + (4/3)π × 27
= 126π + 36π
= 162π cm3
Two hemispheres have the same total volume as one sphere.
Example 3: The Gulab Jamun Question
A gulab jamun is modelled as a cylinder with two hemispherical ends. Its total length is 5 cm and diameter is 2.8 cm. If sugar syrup occupies 30% of its volume, find the approximate syrup volume in 45 gulab jamuns. Use π = 22/7.
Radius = 2.8 ÷ 2
= 1.4 cm
Cylinder length = 5 − 2 × 1.4
= 2.2 cm
Volume of one gulab jamun:
V = πr2h + (4/3)πr3
= (22/7) × (1.4)2 × 2.2
+ (4/3) × (22/7) × (1.4)3
= 13.552 + 11.498666…
= 25.050666… cm3
Syrup volume in 45 gulab jamuns:
= (30/100) × 45 × 25.050666…
= 338.184 cm3
≈ 338.18 mL
Keep intermediate values unrounded so the final answer remains accurate.
Example 4: Capacity of a Cylindrical Container
A cylindrical container has internal radius 7 cm and internal height 20 cm. Find its capacity in litres. Use π = 22/7.
Capacity = πr2h
= (22/7) × 49 × 20
= 3080 cm3
= 3080 ÷ 1000 litres
= 3.08 litres
How to Solve Surface Area and Volume Questions
- Identify whether the question asks for area, volume or capacity.
- Draw a simple sketch and label each component solid.
- Convert diameters to radii and measurements to a common unit.
- Find missing heights, lengths or slant heights.
- For area, list the exposed surfaces that must be measured.
- For volume, add the solid parts or subtract the removed cavity.
- Substitute values, calculate carefully and include the correct units.
Open Cylinder, Closed Cylinder and Tent
A closed cylinder has area 2πrh + 2πr2. An ideal cylinder open at the top but closed at the bottom has area 2πrh + πr2. A conical tent without a floor uses curved surface area πrs for its canvas, unless the question specifies extra material for seams or wastage.
Surface Areas and Volumes Class 10 Mind Map
Use the Surface Areas and Volumes Class 10 mind map in your PDF to connect each shape with its measurements and formulas. Organise your revision into these branches:
- Basic solids: Cube, cuboid, cylinder, cone, sphere and hemisphere.
- Measurements: Radius, diameter, height and slant height.
- Surface area: Curved, lateral, total and exposed surfaces.
- Combined volume: Add component volumes or subtract cavities.
- Applications: Toys, containers, tents and capsule-shaped objects.
- Units: Square units, cubic units, millilitres and litres.
For your 2026 revision, cover the formula chart and recall each formula from the mind map. Then explain why a particular base is included or excluded in a combined-solid question.
Quick Practice Questions with Answers
- A cone has radius 5 cm and height 12 cm. Find its slant height. Answer: 13 cm.
- Find the curved surface area of a hemisphere of radius 7 cm using π = 22/7. Answer: 308 cm2.
- Find the total surface area of the same hemisphere. Answer: 462 cm2.
- Convert 2500 cm3 into litres. Answer: 2.5 litres.
- A cone and cylinder have equal base radii and equal heights. Compare their volumes. Answer: Cone volume : cylinder volume = 1 : 3.
Practise with Related Class 10 Maths Resources
Apply these concepts using the Surface Areas and Volumes Class 10 worksheet with solutions . Attempt the questions independently before checking the working.
Strengthen your understanding of circular measurements with Areas Related to Circles Class 10 notes and mind map . For more revision material, explore our CBSE Class 10 Maths notes collection .
Frequently Asked Questions
What does the Surface Areas and Volumes Class 10 PDF contain?
The PDF contains notes and a mind map. Use the notes to revise concepts and the mind map to organise formulas. The explanations and solved examples on this page provide additional support for applying them.
How do I decide whether to use CSA or TSA?
Identify the surfaces described in the question. Use CSA when only the curved surface is measured. Use TSA when all surfaces of the solid are included. For open or combined objects, list the actual exposed surfaces rather than choosing a formula from one word alone.
Why are joined bases excluded from a combined solid’s surface area?
Joined bases lie inside the combined solid and are not part of its outside boundary. For a cone joined to a hemisphere along equal bases, the exposed area is πrs + 2πr2.
Do we include the base when finding canvas for a conical tent?
For a tent without a floor, use the cone’s curved surface area πrs. Include a circular base only if a floor is specified. Add any allowance for seams or wastage only when the question asks for it.
What is the difference between height and slant height of a cone?
Height is the perpendicular distance from the vertex to the centre of the base. Slant height runs along the sloping surface. Cone volume uses perpendicular height; cone curved surface area uses slant height.
Why is a hemisphere’s TSA different from its CSA?
A hemisphere’s curved area is 2πr2. Its flat circular base adds πr2, giving total surface area 3πr2. Include that base when it forms part of the surface being measured.
How do I find cylinder height in the gulab jamun question?
Subtract the lengths contributed by both hemispherical ends. Each end contributes one radius, so cylinder height or length equals total length − 2r. For total length 5 cm and radius 1.4 cm, it is 2.2 cm.
How do I find the volume of a solid with a cavity?
Subtract the cavity’s volume from the original solid’s volume. For example, if a conical cavity is removed from a cylinder, remaining volume = cylinder volume − cone volume. Use the dimensions of each shape separately.
Does melting and recasting preserve surface area or volume?
Under the usual school assumption that no material is lost, volume remains equal before and after recasting. Surface area generally changes. If one solid forms identical smaller solids, their number equals the original volume divided by the volume of one smaller solid.
How do I convert cubic centimetres into litres?
Divide by 1000. For example, 4500 cm3 = 4.5 litres. To convert cubic centimetres into millilitres, the numerical value stays the same because 1 cm3 = 1 mL.
Are Exercise 12.1 and Exercise 13.1 from different chapters?
Chapter and exercise numbering can differ between textbook editions. Identify the chapter by its name and check your prescribed book before using exercise solutions. Match the complete question rather than relying only on its number.
What are the most common mistakes in this chapter?
Common errors include using diameter as radius, using slant height in cone volume, counting hidden contact surfaces, treating total length as cylinder length and mixing units. Check these points before reviewing your arithmetic.
How should I revise Surface Areas and Volumes Class 10?
Revise the formula chart, study one combined-area example and one combined-volume example, then practise capacity and unit conversion questions. Use the mind map for recall and the worksheet for practice. Explain your choice of surfaces or volumes before calculating.
Download
