Surface Areas & Volumes Class 10: Formula Sheet

Q: What is the difference between curved surface area and total surface area?

Curved surface area (CSA) includes only the curved part of a solid, such as the side of a cylinder or cone, excluding flat faces. Total surface area (TSA) includes the curved surface plus all flat faces, such as the circular top and bottom of a cylinder.

Q: How do you find the surface area of a combination of two solids?

For a combination of solids, such as a cone mounted on a hemisphere, you add only the exposed curved surfaces of each part — not their full total surface areas. The flat face where the two solids join is hidden and must be excluded from the total.

Q: Is volume conserved when a solid is melted and recast into another shape?

Yes. When a solid is melted and recast into a different shape, its volume stays exactly the same, since the same amount of material is used. Surface area, however, is not conserved and will generally be different for the new shape.

Q: What is a frustum of a cone?

A frustum is the solid that remains when the top portion of a cone is sliced off by a plane parallel to its base. It has two circular faces of different radii, and is commonly seen in shapes like buckets and lampshades.

Q: What are the most common mistakes in Class 10 Surface Areas and Volumes?

Common mistakes include adding the full total surface areas of combined solids instead of excluding the joined face, confusing curved surface area with total surface area, forgetting to calculate the slant height before using cone or frustum formulas, and mixing up units, such as forgetting to convert cubic centimetres to litres in capacity problems.

Surface Areas and Volumes Class 10 formula sheet with important area and volume formulas

Surface Areas and Volumes — Class 10 Formula Sheet & Examples (2026)

📅 Updated Sep 15, 2026 ⏱ 14 min read 📘 CBSE / NCERT Class 10, Mensuration ✍️ MyTestSeries Expert Team

Surface Areas and Volumes Class 10 — formulas for cylinder, cone, sphere and combination of solids explained by MyTestSeries

Surface Areas and Volumes doesn't ask you to learn new mathematics as much as it asks you to organise a lot of formulas correctly — one set for the cube, another for the cylinder, another for the cone, and so on. The real challenge shows up when two solids are joined together, or when one shape is melted and reshaped into another, because that's where students who know every individual formula still make avoidable mistakes.

This formula sheet lays out every solid-shape formula in one place, explains exactly how combination-of-solids and melting-and-recasting problems work, and walks through six solved examples — including the classic toy (cone-on-hemisphere) and bucket (frustum) problems — followed by the mistakes that quietly cost marks even when the underlying formula is known perfectly well.

7Solid Shapes Covered

6Solved Examples

7Common Mistakes

1Quick Formula Table

📋 Table of Contents

Cube, Cuboid, Cylinder, Cone & Sphere Formulas
Combination of Solids — The Key Idea
Melting & Recasting
Frustum of a Cone
6 Fully Solved Examples
Common Mistakes Students Make
Quick Formula Recap Table
How to Revise This Chapter Effectively
Frequently Asked Questions
Related Test Series & Practice Sets

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1. Cube, Cuboid, Cylinder, Cone & Sphere Formulas

Solid	Curved/Lateral Surface Area	Total Surface Area	Volume
Cube (side a)	—	6a²	a³
Cuboid (l, b, h)	—	2(lb+bh+hl)	lbh
Cylinder (r, h)	2πrh	2πr(r+h)	πr²h
Cone (r, h, slant l)	πrl	πr(r+l)	(1/3)πr²h
Sphere (r)	—	4πr²	(4/3)πr³
Hemisphere (r)	2πr²	3πr²	(2/3)πr³

For a cone: slant height l = √(r² + h²) This comes directly from the Pythagoras theorem applied to the cone's radius, height, and slant — always calculate l before using any cone surface area formula, unless it's already given.

2. Combination of Solids — The Key Idea

💡 The one rule that matters most in this chapter: When two solids are joined (like a cone on top of a hemisphere), the surface area of the combined shape is the sum of only the exposed curved surfaces — never the sum of their full total surface areas. The flat face where they join is hidden inside the new solid and must be left out entirely.

Volume, on the other hand, behaves more simply: the volume of a combined solid is always just the sum of the individual volumes, since volume measures the space occupied and joining two solids doesn't remove any of that space.

Combination-of-solids questions are where marks are usually won or lost in this chapter. Practise them with the 7-Day Demo Test Series — instant, explained solutions after every attempt.

3. Melting & Recasting

Volume before melting = Volume after recasting Only volume is conserved — surface area is not, and will generally be different once the material takes a new shape.

4. Frustum of a Cone

A frustum is what's left of a cone once its top portion is sliced off by a plane parallel to the base — the shape seen in buckets, lampshades, and drinking glasses.

Slant height: l = √(h² + (r₁−r₂)²)
CSA = π(r₁+r₂)l TSA = π[(r₁+r₂)l + r₁² + r₂²]
Volume = (1/3)πh(r₁² + r₂² + r₁r₂) r₁ and r₂ are the radii of the two circular faces (r₁ > r₂), and h is the vertical height of the frustum.

5. Six Fully Solved Examples

Example 1 — Volume of a cylinder

Find the volume of a cylinder with radius 7 cm and height 10 cm. (Use π = 22/7.)

V = πr²h = (22/7) × 7² × 10 = (22/7) × 49 × 10 = 22 × 7 × 10 = 1540 cm³

Example 2 — Total surface area of a cube

Find the total surface area of a cube with side 5 cm.

TSA = 6a² = 6 × 5² = 6 × 25 = 150 cm²

Example 3 — Combination of solids (surface area)

A toy is a cone mounted on a hemisphere, both of radius 3.5 cm, with total height 15.5 cm. Find its total surface area.

Height of the cone alone = 15.5 − 3.5 = 12 cm (since the hemisphere's radius accounts for the remaining height).
Slant height: l = √(3.5² + 12²) = √(12.25+144) = √156.25 = 12.5 cm
CSA of cone = πrl = (22/7)(3.5)(12.5) = 137.5 cm². CSA of hemisphere = 2πr² = 2(22/7)(3.5²) = 77 cm².
Total surface area = 137.5 + 77 = 214.5 cm² (the flat circular base where they join is hidden, so it's excluded)

Example 4 — Combination of solids (volume)

For the same toy in Example 3, find its total volume.

Volume of cone = (1/3)πr²h = (1/3)(22/7)(3.5²)(12) = 154 cm³
Volume of hemisphere = (2/3)πr³ = (2/3)(22/7)(3.5³) ≈ 89.83 cm³
Total volume = 154 + 89.83 ≈ 243.83 cm³ (unlike surface area, volumes simply add together)

Example 5 — Melting and recasting

A metallic sphere of radius 4.2 cm is melted and recast into a cylinder of radius 6 cm. Find the height of the cylinder.

Volume of sphere = (4/3)πr³ = (4/3)(22/7)(4.2)³ ≈ 310.46 cm³
This becomes the cylinder's volume: πr²h = 310.46 → (22/7)(6²)h = 310.46 → 113.14h = 310.46
h = 310.46/113.14 ≈ 2.74 cm

Example 6 — Frustum (capacity of a bucket)

A bucket is in the shape of a frustum with top radius 15 cm, bottom radius 5 cm, and height 24 cm. Find its capacity.

Volume = (1/3)πh(r₁²+r₂²+r₁r₂) = (1/3)(22/7)(24)(225+25+75)
= (1/3)(22/7)(24)(325) = (22×24×325)/(3×7) = 171600/21 ≈ 8171.4 cm³
Since 1000 cm³ = 1 litre, capacity ≈ 8171.4 cm³ (≈ 8.17 litres)

6. Common Mistakes Students Make

⚠️ Mistake 1 — Adding full total surface areas of combined solids. This always overcounts — the joined flat face is hidden and must be excluded, as shown in Example 3.

⚠️ Mistake 2 — Confusing CSA with TSA. A cylinder's CSA excludes its two circular ends; its TSA includes them. Using the wrong one is one of the most frequent errors in this chapter.

⚠️ Mistake 3 — Using the cone's height instead of its slant height. CSA and TSA of a cone need the slant height l = √(r²+h²), not the vertical height h — mixing these up is an extremely common slip.

⚠️ Mistake 4 — Assuming surface area is conserved in melting-and-recasting problems. Only volume stays the same; surface area almost always changes when the shape changes.

⚠️ Mistake 5 — Using diameter instead of radius. Many problems state the diameter directly — forgetting to halve it before substituting into a formula is a simple but costly slip.

⚠️ Mistake 6 — Arithmetic errors with π = 22/7. Not simplifying fractions before the final multiplication (for example, cancelling a 7 in the denominator against a multiple of 7 in the numerator) leads to unnecessarily messy, error-prone calculations.

⚠️ Mistake 7 — Forgetting unit conversions. Capacity problems often expect litres, not cm³ (1000 cm³ = 1 litre) — and mixed units (cm and m in the same problem) must be converted before any formula is applied.

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7. Quick Formula Recap Table

Solid	Volume
Cube	a³
Cuboid	lbh
Cylinder	πr²h
Cone	(1/3)πr²h
Sphere	(4/3)πr³
Hemisphere	(2/3)πr³
Frustum	(1/3)πh(r₁²+r₂²+r₁r₂)

Formulas stick when you use them. Get 7 full days of Surface Areas & Volumes and other chapter tests with step-by-step solutions — for just ₹10. Try the ₹10 Demo →

8. How to Revise This Chapter Effectively

📖 Build a formula sheet organised by shape, not by mixing area and volume together. Group everything about the cylinder in one row, the cone in another — this mirrors the recap table above and makes recall faster under exam pressure.

✍️ For every combination-of-solids question, physically identify the hidden face before writing anything. Sketch the shape, shade the joined face, and only add up what's left visible.

🧮 Practise slant height calculations as a separate first step. Get into the habit of computing l = √(r²+h²) before touching any cone or frustum surface area formula.

🎯 Double-check units before submitting a final answer. If the question mentions litres, capacity, or mixes cm and metres, do the conversion early — not as an afterthought.

🔁 Revisit this page before your unit test. Bookmark it, and use the Quick Formula Recap Table (Section 7) as your final five-minute revision before the exam.

📚 Also Useful on MyTestSeries

🌐 Trusted External References

NCERT Official Website — download the official Class 10 Maths textbook
CBSE Academic — official syllabus & sample papers
Frustum — Wikipedia — broader mathematical background
Volume — Wikipedia — general theory and other solid formulas

9. Frequently Asked Questions

What is the difference between curved surface area and total surface area?

CSA includes only the curved part of a solid. TSA includes the curved surface plus all flat faces — such as the circular ends of a cylinder.

How do you find the surface area of a combination of two solids?

Add only the exposed curved surfaces of each part — exclude the flat face where the two solids are joined, since it's hidden inside the new shape.

Is volume conserved when a solid is melted and recast into another shape?

Yes, volume stays exactly the same. Surface area is not conserved and will generally differ in the new shape.

What is a frustum of a cone?

The solid left over when the top of a cone is sliced off by a plane parallel to its base — seen in buckets, lampshades, and similar shapes.

What are the most common mistakes in Class 10 Surface Areas and Volumes?

Adding full total surface areas of combined solids instead of excluding the joined face, confusing CSA with TSA, forgetting to calculate slant height, and mixing up units like cm³ and litres.

Conclusion: Organisation Beats Memorisation

This chapter has more formulas than most, but very little genuinely new mathematics — the real skill is applying the right formula to the right shape and, critically, correctly handling what happens when shapes are combined or reshaped. Once the CSA-vs-TSA distinction and the "exclude the joined face" rule are automatic, this becomes one of the more reliably scorable chapters in the whole syllabus.

Rework the six solved examples above without checking the answers first, always sketch combination-of-solids problems before calculating anything, and double-check your units at the very end. That combination — organise, sketch, verify units — is what turns a long formula list into consistent exam marks.

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Tags: Surface Areas and Volumes Class 10, Combination of Solids, Melting and Recasting, Frustum of a Cone, CBSE 2026, NCERT Class 10 Maths Mensuration

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