Chapter 7 of 9

78%

Tamil Nadu (TNBSE)Class 9 Mathematics★ 6-10 marks in the TN Class 9 annual exam through MCQs and area/surface-area/volume problems

Mensuration

Samacheer Kalvi (TN State Board) Class 9 Maths Chapter 7, Mensuration. Heron's formula for the area of a triangle, finding the area of a quadrilateral using Heron's formula, and the surface area and volume of a cube and a cuboid. Notes with formulas, worked examples, common mistakes, book-back practice and quick revision.

⏱ 18 min readMedium difficulty✓ Free · NCERT-aligned

Worksheet

Podcast

Listen as a conversation

A teacher & a student talk through the chapter — in your language.

Mind Map

The chapter at a glance

Every concept and exam-favourite branch — revisable in 5 minutes.

…

Study Guide

Night-before notes

Key points, glossary and the questions most likely to show up.

…

VocabularyBuild your vocabularyLexiGenome mines the key words and schedules them with spaced repetition.Mine words →

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

1Apply Heron's formula for the area of a triangle
2Find the area of a quadrilateral by splitting into triangles
3Compute surface area and volume of a cube
4Compute surface area and volume of a cuboid
5Find the diagonal of a cube and cuboid

💡

Why this chapter matters

Mensuration applies formulas to real shapes — fields, boxes and rooms. Heron's formula and the cube/cuboid surface area and volume are reliable, formula-driven marks in the TN Class 9 exam.

Before you start — revise these

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

🔗

Class 8 — Area and volume

Perimeter, area and basic solids.

Mensuration — Class 9 Maths (Samacheer Kalvi)

TN State Board (Samacheer Kalvi) Class 9 Mathematics, Chapter 7. Areas by Heron's formula and the surface area and volume of cubes and cuboids.

1. About this chapter

This chapter covers Heron's formula for the area of a triangle (and quadrilaterals), and the surface area and volume of a cube and a cuboid.

2. Heron's formula

For a triangle with sides a, b, c and semi-perimeter s = (a + b + c)/2: Area = √[ s(s − a)(s − b)(s − c) ].
Area of a quadrilateral: split it into two triangles using a diagonal and add their areas (each found by Heron's formula).

3. Cube and cuboid

Solid	Total Surface Area	Volume
Cube (side a)	6a²	a³
Cuboid (l, b, h)	2(lb + bh + hl)	l b h

Diagonal of a cuboid = √(l² + b² + h²); diagonal of a cube = a√3.

4. Worked examples

Example 1. Find the area of a triangle with sides 13, 14, 15 cm. s = (13 + 14 + 15)/2 = 21; Area = √[21(21−13)(21−14)(21−15)] = √[21×8×7×6] = √7056 = 84 cm².

Example 2. Find the volume and surface area of a cube of side 5 cm. V = a³ = 125 cm³; TSA = 6a² = 6(25) = 150 cm².

Example 3. Find the volume of a cuboid 8 × 5 × 3 cm. V = lbh = 8 × 5 × 3 = 120 cm³.

5. Common mistakes

Mistake: Forgetting the semi-perimeter in Heron's formula. Fix: Use s = (a + b + c)/2 inside √[s(s−a)(s−b)(s−c)].
Mistake: Using 4a² for a cube's surface area. Fix: A cube has 6 faces → TSA = 6a².
Mistake: Mixing area and volume units. Fix: Area is in cm², volume in cm³.

6. Practice (book-back style)

Write Heron's formula.
Find the area of a triangle with sides 3, 4, 5 cm.
Find the volume of a cube of side 4 cm.
Find the total surface area of a cuboid 6 × 4 × 2 cm.
Find the diagonal of a cube of side 2 cm.

7. Answer key

Area = √[s(s − a)(s − b)(s − c)], s = (a + b + c)/2.
s = 6; Area = √[6(3)(2)(1)] = √36 = 6 cm².
V = 4³ = 64 cm³.
TSA = 2(6×4 + 4×2 + 2×6) = 2(24 + 8 + 12) = 88 cm².
Diagonal = a√3 = 2√3 cm.

8. Quick revision

Chapter 7 · Heron's formula, cube and cuboid.
Heron: Area = √[s(s−a)(s−b)(s−c)], s = (a+b+c)/2.
Quadrilateral area = sum of two triangle areas (split by a diagonal).
Cube: TSA 6a², V a³; Cuboid: TSA 2(lb+bh+hl), V lbh.
Cuboid diagonal = √(l²+b²+h²); cube diagonal = a√3.

Key formulas & results

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

Heron's formula

Area = √[s(s − a)(s − b)(s − c)]

s = (a + b + c)/2.

Cube

TSA = 6a², V = a³

Diagonal = a√3.

Cuboid

TSA = 2(lb + bh + hl), V = lbh

Diagonal = √(l² + b² + h²).

⚠️

Common mistakes & fixes

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

WATCH OUT

✗ Forgetting the semi-perimeter in Heron's formula

✓ Use s = (a + b + c)/2 inside the square root.

WATCH OUT

✗ Using 4a² for a cube's surface area

✓ A cube has 6 faces → TSA = 6a².

WATCH OUT

✗ Mixing area and volume units

✓ Area is in cm², volume in cm³.

NCERT exercises (with solutions)

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

Multiple choice questions

MCQs on Heron's formula, cube and cuboid.

Questions

Area problems

Triangle and quadrilateral areas by Heron's formula.

Questions

Surface area / volume

Cube and cuboid surface area and volume.

Questions

Practice problems

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

Q1EASY· Recall

Write Heron's formula.

▶ Show solution

Area = √[s(s − a)(s − b)(s − c)], where s = (a + b + c)/2.

Q2EASY· Numerical

Find the area of a triangle with sides 3, 4, 5 cm.

▶ Show solution

s = 6; Area = √[6(3)(2)(1)] = √36 = 6 cm².

Q3EASY· Numerical

Find the volume of a cube of side 4 cm.

▶ Show solution

V = 4³ = 64 cm³.

Q4MEDIUM· Numerical

Find the total surface area of a cuboid 6 × 4 × 2 cm.

▶ Show solution

TSA = 2(24 + 8 + 12) = 88 cm².

Q5HARD· Numerical

Find the area of a triangle with sides 13, 14, 15 cm.

▶ Show solution

s = 21; Area = √[21×8×7×6] = √7056 = 84 cm².

Q6EASY· Numerical

Find the diagonal of a cube of side 2 cm.

▶ Show solution

Diagonal = a√3 = 2√3 cm.

5-minute revision

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

•Chapter 7 of Samacheer Kalvi Class 9 Mathematics.
•Heron: Area = √[s(s−a)(s−b)(s−c)], s = (a+b+c)/2.
•Quadrilateral area = sum of two triangle areas.
•Cube: TSA 6a², V a³; Cuboid: TSA 2(lb+bh+hl), V lbh.
•Cuboid diagonal = √(l²+b²+h²); cube diagonal = a√3.
•Area in cm², volume in cm³.

Tamil Nadu (TNBSE) marks blueprint

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

Typical chapter weightage: 6-10 marks across MCQ and area/volume problems

Question type	Marks each	Typical count	What it tests
MCQ	1	1-2	Formulas and units
Area	2-3	1-2	Heron's formula
Surface Area / Volume	2-3	1	Cube and cuboid

Prep strategy

Memorise Heron's formula and the semi-perimeter
Learn cube and cuboid formulas
Split quadrilaterals into triangles
Keep area and volume units distinct

Where this shows up in the real world

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

Land measurement

Heron's formula finds the area of triangular and quadrilateral plots.

Packaging

Cuboid surface area and volume size boxes and cartons.

Construction

Volumes estimate concrete and material quantities.

Exam strategy

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

Compute s first in Heron's problems
Use 6 faces for a cube's surface area
Split quadrilaterals along a diagonal
Label units (cm² vs cm³)

Going beyond the textbook

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

Derive the area of an equilateral triangle from Heron's formula.
Find the longest rod that fits in a given cuboidal room.

Where else this chapter is tested

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

TN Class 9 Annual ExamHigh

Foundation / NTSE MathematicsMedium

School unit testsHigh

AI helpers for this chapter

Generate more practice on the fly, or have the chapter explained at a different level.

⚡

Generate 5 more questions

Fresh MCQs + short-answer items based on this chapter, plus answer keys.

🧠

Explain at a different level

Re-explain the chapter at the level you actually want.

📝 My notes

Saved on this device — sign in to sync · how this works

0/600

Questions students ask

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

When you know all three sides of a triangle but not its height — Heron's formula gives the area directly from the sides.

Draw a diagonal to split it into two triangles, find each area by Heron's formula, and add them.

Practice more, ask doubts, find a tutor

Tamil Nadu (TNBSE) Class 9 mock tests

Tamil Nadu (TNBSE) Class 9 PYQs

Ask in Q&A

Mathematics flashcards

Related chapters

Students who read Mensuration usually read these next.

Set Language

Samacheer Kalvi (TN State Board) Class 9 Maths Chapter 1, Set Language. Sets and their representation, types of sets, set operations (union, intersection, difference, complement), properties, De Morgan's laws, and cardinality with Venn diagrams. Notes with formulas, worked examples, common mistakes, book-back practice and quick revision.

Real Numbers

Samacheer Kalvi (TN State Board) Class 9 Maths Chapter 2, Real Numbers. Rational and irrational numbers, decimal representation, surds and operations on surds, rationalising the denominator, and scientific notation. Notes with formulas, worked examples, common mistakes, book-back practice and quick revision.

Algebra

Samacheer Kalvi (TN State Board) Class 9 Maths Chapter 3, Algebra. Polynomials, algebraic identities and factorisation, the remainder and factor theorems, GCD and LCM of polynomials, simultaneous linear equations, and linear inequations in two variables. Notes with formulas, worked examples, common mistakes, book-back practice and quick revision.

Geometry

Samacheer Kalvi (TN State Board) Class 9 Maths Chapter 4, Geometry. Triangles and congruence, the mid-point theorem, properties of quadrilaterals and parallelograms, circle theorems (chords and cyclic quadrilaterals), and constructions of triangle centres. Notes with theorems, worked examples, common mistakes, book-back practice and quick revision.

🧑‍🏫