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
Trapezium
- Area = 1/2 × (sum of parallel sides) × height
- Area = 1/2 × (a + b) × h where a and b are parallel sides
Circle
Key Terms:
- Radius (r): Distance from centre to circumference
- Diameter (d): 2r
- Circumference: 2πr or πd
- Area: πr²
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°.
