SOLIDS • EXPOSED SURFACES • CAPACITY
Surface Areas and Volumes
Worksheet with Online Quiz
Recognise the solid, choose the measurement and count only the surfaces that matter. Explore cylinders, cones, hemispheres and combined shapes.
Understand the Solid
Identify radius, height, slant height and shared faces.
Explore the diagrams →Count Exposed Surfaces
Choose curved, total or exposed area for the situation.
Try the activity labs →Measure Space and Capacity
Combine volumes and convert units with confidence.
Solve the practical cases →Surface Areas and Volumes Online Quiz
Question 1 of 20 • Easy • Answered 0/20
Which unit is appropriate for the volume of a solid?
Quiz instructions
Attempt 10 easy, five average and five HOTS questions in 20 minutes. The timer starts with your first answer or question navigation. Selecting an option opens the next question. Use Previous or the numbered buttons to revisit answers. One mark per correct answer and no negative marking. Solutions appear after submission or timeout.
Interactive Surface Area and Volume Games
Touch the Solid
Tap a marked part or use the labelled buttons.
Select the cylinder’s radius.
Choose your game
Use the raised selector to explore solid parts, adjust cylinder height, match formulas, sort measurements or build solutions. Each game contains five tasks.
Solid Geometry Calculation Activities
Answer entry help
Use π=22/7 in the numerical activities. Enter values without units. Fractions, radicals and expressions such as 12π, 12*pi and sqrt(169) are supported. Here π and pi are evaluated using the stated 22/7 convention. Give non-terminating decimals to at least three places.
Extra Practice with Solids and Capacity
Find the Missing Dimension
Rearrange the correct formula. Use π=22/7.
A cylinder of radius 7 cm has volume 1848 cm³. Find its height.
Use π=22/7. Enter fractions or expressions such as 12π, 12*pi or sqrt(169). For decimals, use at least three decimal places.
Explore the practice sets
Find missing dimensions, convert units, estimate covering costs, solve model-design cases and create valid solids. Each set contains five tasks. Entries remain available while this page stays open.
Revision Tips for Students
- Identify each basic solid before choosing a formula.
- Check whether the question asks for area, volume, capacity or cost.
- Use perpendicular height in cone volume and slant height in cone curved area.
- Convert diameter to radius before calculating.
- Exclude internal joining faces from exposed area.
- Add volumes of joined solids whose interiors do not overlap; subtract removed cavities.
- Use internal dimensions for capacity and include only the named surfaces for painting.
- Convert units before substitution and finish with square or cubic units as appropriate.
Teaching Tips for Parents and Teachers
- Ask students to name the solids inside a combined model.
- Use detachable shapes to show why a joining face disappears from exposed area.
- Compare a cone’s vertical height with its sloping edge.
- Use the cylinder slider to explore how height changes volume.
- Ask students to explain which surfaces are included before calculating a cost.
Surface Areas and Volumes Notes Formulas and Solved Questions
This Surface Areas and Volumes Class 10 worksheet develops the link between a solid’s diagram and its measurement. Start with the online quiz, then practise formula selection, hidden faces, combinations of solids and capacity. Use the existing worksheet download option on this page for written revision.
Difference Between CSA TSA and Volume
Curved surface area, or CSA, measures only a solid’s curved boundary. Lateral surface area, or LSA, measures side faces of solids such as cuboids. Total surface area, or TSA, includes all boundary faces of an individual solid. Volume measures the space occupied. Capacity is the internal volume a vessel can hold, so use internal dimensions when wall thickness matters.
Surface Areas and Volumes Class 10 Formula Sheet
Here r is radius, h perpendicular height, l cone slant height, a cube edge, and L, B, H are cuboid dimensions.
| Solid | Curved or lateral area | Total area | Volume |
|---|---|---|---|
| Cube | 4a² | 6a² | a³ |
| Cuboid | 2H(L+B) | 2(LB+BH+HL) | LBH |
| Closed cylinder | 2πrh | 2πr(h+r) | πr²h |
| Right circular cone | πrl | πr(l+r) | (1/3)πr²h |
| Sphere | 4πr² | 4πr² | (4/3)πr³ |
| Solid hemisphere | 2πr² | 3πr² | (2/3)πr³ |
For a right circular cone, l²=r²+h². Find l with Pythagoras when only r and h are given.
Surface Area of a Combination of Solids
List the surfaces still exposed after the pieces are joined. A cone on a cylinder of equal radius hides the shared circular face. Its closed solid has exposed area 2πrh(cylinder)+πrl+πr², including the cylinder’s bottom. For a tent without a floor, omit that bottom. A hemisphere joined to a cone along their equal bases leaves only their curved surfaces exposed.
Volume of a Combination of Solids
Split the object into familiar solids. Add their volumes when they meet only along a face. For a cavity, subtract the removed volume. If the total height includes a hemispherical cap, the cylinder height is total height minus one radius. A cone on a hemisphere follows the same height logic.
Surface Areas and Volumes Class 10 Important Questions
Practise formula selection, cone slant height, exposed areas, combined volumes, unknown dimensions, capacity, unit conversions and costs. Useful models include a cylinder with a cone roof, a cone–hemisphere toy and joined cubes. The word “important” describes revision coverage and does not predict a future paper.
Surface Areas and Volumes Class 10 Examples with Solutions
Choose the correct cylinder surface
A thin cylindrical vessel has radius 7 cm and height 15 cm. It is open at the top and closed at the bottom. Find material area, ignoring thickness. Use π=22/7.
Area=2πrh+πr²
Area=2×(22/7)×7×15+(22/7)×7²
Area=660+154
Area=814 cm²
Find cone slant height before curved area
A cone has radius 12 cm and perpendicular height 16 cm. Find its CSA. Use π=22/7.
l²=r²+h²
l²=12²+16²
l²=400
l=20 cm
CSA=πrl
CSA=(22/7)×12×20
CSA=5280/7 cm²
Find combined volume
A solid cylinder of radius 3 cm and height 10 cm is capped by a solid hemisphere of the same radius. Find volume. Use π=22/7.
V=πr²h+(2/3)πr³
V=90π+18π
V=108π
V=2376/7 cm³
The full height is 10+3=13 cm, but only 10 cm belongs in the cylinder formula.
Surface Areas and Volumes Class 10 Case-Based Questions
Plan a cylindrical tent with a cone roof
A tent has a cylindrical wall of radius 7 m and height 3 m. The roof has perpendicular height 24 m. No floor is covered. Use π=22/7.
Find the roof slant height.
l=√(7²+24²)
l=25 m
Find the canvas area before allowances.
Area=2πrh+πrl
Area=132+550
Area=682 m²
If 18 m² is added for stitching and canvas costs ₹40 per m², find the cost.
Canvas bought=682+18
Canvas bought=700 m²
Cost=700×40
Cost=₹28000
Surface Areas and Volumes Class 10 Extra Questions with Answers
Use π=22/7 where needed.
- A cylinder has radius 7 cm and height 13 cm. Find volume.
View solution
V=(22/7)×7²×13
V=2002 cm³
- A cone has radius 6 cm and height 8 cm. Find slant height.
View solution
l=√(6²+8²)
l=10 cm
- A solid hemisphere has radius 14 cm. Find TSA.
View solution
TSA=3×(22/7)×14²
TSA=1848 cm²
- Two cubes of edge 6 cm are joined along a complete face. Find exposed area.
View solution
Area=12a²−2a²
Area=10×6²
Area=360 cm²
- Convert 7350 cm³ to litres.
View solution
Capacity=7350/1000
Capacity=7.35 L
How to Practise NCERT Surface Areas and Volumes Exercises
Begin each problem with a labelled sketch and an explicit list of included surfaces or volumes. For exposed area, mark the hidden joining faces. For volume, use the correct height of each component. Match exercise numbers to your textbook edition and check the exact wording before using a solution. These examples are original practice, not a complete reproduction of NCERT exercises.
Current Syllabus and Official Sample-Paper Update
The official CBSE Mathematics syllabus for 2026–27 lists surface areas and volumes of combinations of two solids drawn from cubes, cuboids, spheres, hemispheres and right circular cylinders or cones. The current official Mathematics Standard sample paper includes joined cubes, a cylinder–cone tent and a cone–hemisphere toy. Practise hidden faces, component heights, combined volumes and cost calculations. These are verified sample-paper themes, not a claim that particular questions are guaranteed to repeat.
Common Surface Area and Volume Mistakes
Avoid using slant height in cone volume, using perpendicular height in cone CSA, counting a joining disk as exposed, including a floor when none is covered, using outer dimensions for internal capacity, or applying a length conversion directly to square or cubic units.
Related Maths Worksheets and Revision
Surface Areas and Volumes Frequently Asked Questions
What is the difference between CSA TSA and volume?
CSA measures the curved part of a boundary. TSA includes every boundary face of an individual solid. Volume measures occupied space. Surface areas use square units; volumes use cubic units.
When should I use curved surface area instead of total surface area?
Use CSA when only the curved side is covered or painted, such as a conical tent without a floor. For a closed cylinder, include both circular bases. Follow the surfaces explicitly named in the question.
How do I find the surface area of combined solids?
List the exposed surfaces after joining the parts. Exclude their common internal faces. If adding the two original TSAs, subtract both copies of a completely hidden shared face.
What is the difference between cone height and slant height?
Perpendicular height runs from apex to base centre. Slant height runs from apex to the rim. In a right circular cone, l²=r²+h². Use l in curved area and h in volume.
Why is hemisphere CSA 2πr² but TSA 3πr²?
The curved half of a sphere has area 2πr². Adding the flat circular base πr² gives 3πr². A hemispherical dome’s curved cover uses CSA.
How do I find the volume of a combination of solids?
Add the component volumes when their interiors do not overlap. Subtract any removed cavity. Use the dimensions of each component rather than the full combined height in every formula.
How do I find cylinder height when there is a hemispherical cap?
Subtract the hemisphere’s radius from the full height. A hemisphere contributes height r, not diameter 2r. The remaining height belongs to the cylinder.
How do I calculate the capacity of a vessel?
Find the volume of its internal space using internal dimensions. Ignore thickness only when the question allows it. For a combined vessel, add the internal component volumes.
How do I convert cm³ to litres and m³?
1000 cm³=1 L and 1000 L=1 m³. Therefore 1000000 cm³=1 m³. Convert each dimension to a consistent unit before calculating.
Which Surface Areas and Volumes questions are important for exams?
Cover formula selection, cone slant height, combined surface areas and volumes, hidden faces, capacity, missing dimensions and costs. Use official sample papers for verified exam-style examples.
How do I solve painting or canvas cost questions?
Identify the surfaces being covered, calculate their area, add any stated allowance and multiply by the rate per square unit. An open tent does not need floor canvas unless the question says otherwise.
Do surface area and volume change by the same factor when dimensions double?
No. For similar solids with every length doubled, surface area becomes four times as large and volume becomes eight times as large. The scale factors are k² and k³.
Can You See Which Faces Stay Exposed?
Identify the parts, choose the formula and test your understanding.
Start the Online Quiz
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