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Tamil Nadu (TNBSE)Class 10 Mathematics★ 8-12 marks in the TN SSLC Class 10 Maths public exam through MCQs and surface-area/volume problems

Mensuration

Samacheer Kalvi (TN State Board) Class 10 Maths Chapter 7, Mensuration. Surface area and volume of cylinders, cones, spheres and hemispheres, the frustum of a cone, combinations of solids, and conversion of solids with no change in volume. Notes with formulas, worked examples, common mistakes, book-back practice and quick revision.

⏱ 22 min readMedium difficulty✓ Free · NCERT-aligned

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By the end of this chapter you'll be able to…

1Find surface area and volume of cylinder, cone, sphere and hemisphere
2Compute the slant height of a cone and frustum
3Solve frustum surface-area and volume problems
4Handle combinations of solids
5Apply volume conservation in conversion of solids

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Why this chapter matters

Mensuration applies geometry to real solids — tanks, cones, domes and buckets. Surface area, volume, frustum and conversion problems are heavily weighted and reliably scoring in the TN SSLC exam.

Before you start — revise these

A 5-minute refresher here will save you 30 minutes of confusion below.

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Class 9 — Surface areas and volumes

Basic solids and their formulas.

Mensuration — Class 10 Maths (Samacheer Kalvi)

TN State Board (Samacheer Kalvi) Class 10 Mathematics, Chapter 7. Surface areas and volumes of solids.

1. About this chapter

This chapter covers the surface area and volume of the cylinder, cone, sphere, hemisphere and frustum, plus combinations of solids and conversion of solids (volume stays the same).

2. Standard solids

Solid	Curved/Lateral SA	Total SA	Volume
Cylinder	2πrh	2πr(h + r)	πr²h
Cone	πrl	πr(l + r)	⅓πr²h
Sphere	4πr²	4πr²	(4/3)πr³
Hemisphere	2πr²	3πr²	(2/3)πr³

Cone slant height: l = √(r² + h²).

3. Frustum of a cone

A frustum is formed when a cone is cut by a plane parallel to its base.
Slant height l = √(h² + (R − r)²).
CSA = πl(R + r); TSA = πl(R + r) + πR² + πr²; Volume = ⅓πh(R² + Rr + r²).

4. Combination and conversion of solids

Combination: add the surface areas/volumes of the parts (e.g., a cone on a hemisphere).
Conversion: when one solid is melted and recast into another, the volume is conserved (volume before = volume after).

5. Worked examples

Example 1. Find the volume of a cylinder with r = 7 cm, h = 10 cm. (π = 22/7) V = πr²h = (22/7)(49)(10) = 1540 cm³.

Example 2. A cone has r = 3 cm, h = 4 cm. Find its slant height and CSA. l = √(3² + 4²) = 5 cm; CSA = πrl = (22/7)(3)(5) = 47.1 cm² (approx).

Example 3. A metallic sphere of radius 3 cm is melted and recast into a cylinder of radius 3 cm. Find its height. Volume conserved: (4/3)π(3)³ = π(3)²h → 36π = 9πh → h = 4 cm.

6. Common mistakes

Mistake: Using diameter instead of radius. Fix: All formulas use the radius (r = d/2).
Mistake: Forgetting the base areas in TSA of a frustum. Fix: TSA = πl(R + r) + πR² + πr².
Mistake: Changing the volume during conversion. Fix: Volume is conserved when a solid is melted and recast.

7. Practice (book-back style)

Write the curved surface area of a cone.
Find the volume of a sphere of radius 6 cm (in terms of π).
A cylinder has r = 5 cm, h = 14 cm. Find its CSA. (π = 22/7)
Write the volume formula for the frustum of a cone.
A cone of radius 6 cm and height 8 cm — find its slant height.

8. Answer key

CSA of a cone = πrl.
V = (4/3)π(6)³ = 288π cm³.
CSA = 2πrh = 2(22/7)(5)(14) = 440 cm².
Volume = ⅓πh(R² + Rr + r²).
l = √(6² + 8²) = √100 = 10 cm.

9. Quick revision

Chapter 7 · surface area and volume of solids.
Cylinder: CSA 2πrh, V πr²h; Cone: CSA πrl, V ⅓πr²h, l = √(r²+h²).
Sphere: SA 4πr², V (4/3)πr³; Hemisphere: TSA 3πr², V (2/3)πr³.
Frustum: V = ⅓πh(R² + Rr + r²); CSA = πl(R + r).
Conversion of solids conserves volume.

Key formulas & results

Everything you need to memorise, in one card. Screenshot this for revision.

Cylinder

CSA 2πrh, V πr²h

TSA = 2πr(h + r).

Cone

CSA πrl, V ⅓πr²h, l = √(r² + h²)

TSA = πr(l + r).

Sphere / hemisphere

Sphere V (4/3)πr³; Hemisphere V (2/3)πr³, TSA 3πr²

Sphere SA 4πr².

Frustum

V = ⅓πh(R² + Rr + r²); CSA = πl(R + r)

l = √(h² + (R − r)²).

Conversion of solids

volume before = volume after

Melting and recasting conserves volume.

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Common mistakes & fixes

These are the exact errors that cost students marks in board exams. Read them once, save yourself the trouble.

WATCH OUT

✗ Using diameter instead of radius

✓ All formulas use the radius (r = d/2).

WATCH OUT

✗ Forgetting the base areas in TSA of a frustum

✓ TSA = πl(R + r) + πR² + πr².

WATCH OUT

✗ Changing the volume during conversion

✓ Volume is conserved when a solid is melted and recast.

NCERT exercises (with solutions)

Every NCERT exercise from this chapter — what it covers and how many questions to expect.

Multiple choice questions

MCQs on surface areas and volumes.

Questions

Surface-area problems

CSA and TSA of solids and combinations.

Questions

Volume / conversion problems

Volume, frustum and conversion-of-solids sums.

Questions

Practice problems

Try each one yourself before tapping "Show solution". Active recall > rereading.

Q1EASY· Recall

Write the curved surface area of a cone.

▶ Show solution

CSA of a cone = πrl.

Q2EASY· Numerical

Find the volume of a sphere of radius 6 cm (in terms of π).

▶ Show solution

V = (4/3)π(6)³ = 288π cm³.

Q3MEDIUM· Numerical

A cylinder has r = 5 cm, h = 14 cm. Find its CSA (π = 22/7).

▶ Show solution

CSA = 2πrh = 2(22/7)(5)(14) = 440 cm².

Q4EASY· Numerical

A cone of radius 6 cm and height 8 cm — find its slant height.

▶ Show solution

l = √(6² + 8²) = √100 = 10 cm.

Q5HARD· Conversion

A metallic sphere of radius 3 cm is melted and recast into a cylinder of radius 3 cm. Find its height.

▶ Show solution

(4/3)π(3)³ = π(3)²h → 36π = 9πh → h = 4 cm.

Q6EASY· Recall

Write the volume formula for the frustum of a cone.

▶ Show solution

Volume = ⅓πh(R² + Rr + r²).

5-minute revision

The whole chapter, distilled. Read this the night before the exam.

•Chapter 7 of Samacheer Kalvi Class 10 Mathematics.
•Cylinder: CSA 2πrh, V πr²h; Cone: CSA πrl, V ⅓πr²h, l = √(r²+h²).
•Sphere: SA 4πr², V (4/3)πr³; Hemisphere: TSA 3πr², V (2/3)πr³.
•Frustum: V = ⅓πh(R² + Rr + r²); CSA = πl(R + r).
•Combination: add parts; Conversion: volume conserved.
•Use radius (r = d/2) in every formula.

Tamil Nadu (TNBSE) marks blueprint

Where the marks come from in this chapter — so you can plan your prep.

Typical chapter weightage: 8-12 marks across MCQ and surface-area/volume problems

Question type	Marks each	Typical count	What it tests
MCQ	1	1-2	Formulas and units
Surface Area	2-5	1-2	CSA/TSA of solids and combinations
Volume / Conversion	2-5	1-2	Volume, frustum, conversion

Prep strategy

Memorise all solid formulas
Practise frustum and combination problems
Use volume conservation for conversions
Always use radius, not diameter

Where this shows up in the real world

This chapter isn't just an exam topic — it lives in the world around you.

Tanks and containers

Volume formulas size water tanks, cans and buckets.

Manufacturing

Recasting metal uses volume conservation.

Construction

Surface areas estimate paint and material needs.

Exam strategy

Battle-tested tips from teachers and toppers for this chapter.

Write the formula and substitute carefully
Convert diameter to radius first
For combinations, add the right faces only
Use volume = volume for conversions

Going beyond the textbook

For olympiad aspirants and curious learners — topics that build on this chapter.

Derive the frustum volume from two cones.
Find the surface area of a capsule (cylinder with two hemispheres).

Where else this chapter is tested

CBSE board isn't the only one — other exams test this chapter too.

TN SSLC Class 10 Public ExamHigh

Foundation / NTSE MathematicsMedium

School unit testsHigh

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Questions students ask

The real ones — pulled from the Q&A community and tutor sessions.

Set the volume of the original solid equal to the volume of the new solid, since melting and recasting conserves volume, then solve for the unknown dimension.

The portion of a cone left after the top is cut off by a plane parallel to the base — shaped like a bucket.

Practice more, ask doubts, find a tutor

Tamil Nadu (TNBSE) Class 10 mock tests

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