Mensuration - Area and Perimeter

Introduction

Mensuration deals with the measurement of geometric figures. For ICSE Class 9, understanding area and perimeter of plane figures is essential for solving real-world problems.

Plane Figures: Area and Perimeter

Triangle

Type	Formula for Area
General	1/2 × base × height
Equilateral	(√3/4) × side²
Right triangle	1/2 × product of legs

Perimeter = Sum of all sides

<ICSEExample title="Area of Triangle"> Find the area of a triangle with base 12 cm and height 8 cm. <Solution> Area = 1/2 × base × height Area = 1/2 × 12 × 8 = 48 cm² </Solution> </ICSEExample>

Rectangle and Square

Rectangle:

Area = length × breadth
Perimeter = 2(l + b)
Diagonal = √(l² + b²)

Square:

Area = side²
Perimeter = 4 × side
Diagonal = side × √2

Parallelogram

Area = base × height (height is perpendicular to base)
Perimeter = 2(sum of adjacent sides)

Rhombus

Area = 1/2 × d1 × d2 (where d1, d2 are diagonals)
Area = base × height
Perimeter = 4 × side

<ICSEExample title="Area of Rhombus"> Find the area of a rhombus whose diagonals are 10 cm and 14 cm. <Solution> Area = 1/2 × d1 × d2 Area = 1/2 × 10 × 14 = 70 cm² </Solution> </ICSEExample>

Trapezium

Area = 1/2 × (sum of parallel sides) × height
Area = 1/2 × (a + b) × h where a and b are parallel sides

<ICSEExample title="Area of Trapezium"> Find the area of a trapezium with parallel sides 8 cm and 12 cm and height 5 cm. <Solution> Area = 1/2 × (8 + 12) × 5 Area = 1/2 × 20 × 5 = 50 cm² </Solution> </ICSEExample>

Circle

Key Terms:

Radius (r): Distance from centre to circumference
Diameter (d): 2r
Circumference: 2πr or πd
Area: πr²

<ICSEExample title="Circle Calculations"> Find the circumference and area of a circle with radius 7 cm. (Use π = 22/7) <Solution> Circumference = 2πr = 2 × 22/7 × 7 = 44 cm Area = πr² = 22/7 × 7 × 7 = 154 cm² </Solution> </ICSEExample>

Area of a Sector

A sector is the region between two radii and the arc.

Area of sector = (θ/360°) × πr²
Length of arc = (θ/360°) × 2πr

Where θ is the central angle in degrees.

<ICSEExample title="Area of Sector"> Find the area of a sector of a circle with radius 14 cm and central angle 90°. <Solution> Area = (90/360) × π × 14² = 1/4 × 22/7 × 196 = 1/4 × 22 × 28 = 154 cm² </Solution> </ICSEExample>

Area Between Rectangles

The area between two concentric rectangles (a path around a rectangle):

Area = (Outer area) - (Inner area)

<ICSEExample title="Area of Path"> A rectangular garden 30 m by 20 m has a path of width 2 m around it on the outside. Find the area of the path. <Solution> Outer length = 30 + 2 + 2 = 34 m Outer breadth = 20 + 2 + 2 = 24 m Outer area = 34 × 24 = 816 m² Inner area = 30 × 20 = 600 m² Area of path = 816 - 600 = 216 m² </Solution> </ICSEExample>

Common Mistakes With Fixes

Mistake	Correction
Confusing perimeter and area units	Perimeter: linear units (cm), Area: square units (cm²)
Using diameter instead of radius in circle area	Area = πr², NOT πd²
Forgetting to use perpendicular height in triangles	Height must be perpendicular to the base
Confusing circumference formula with area	Circumference = 2πr, Area = πr²

ICSE Exam Focus

Topic	Marks (approx.)	Frequency
Area and perimeter of quadrilaterals	4-5 marks	Very common
Circle area and circumference	3-4 marks	Very common
Area of sector and arc length	4 marks	Common
Composite figures and paths	4-5 marks	Frequently asked

Self-Test

Q1: Find the area of a triangle with sides 13 cm, 14 cm, and 15 cm. (Heron's formula optional, use base and height)

Q2: A rectangular field is 50 m by 40 m. A path of width 2 m runs around it inside. Find the area of the path.

Q3: Find the area of a circle whose circumference is 88 cm.

Q4: A rhombus has area 96 cm² and one diagonal 16 cm. Find the other diagonal.

Q5: Find the area of a sector of a circle with radius 21 cm and angle 60°.

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