ICSE Class 6 Maths Chapter 23 Perimeter and Area of Plane Figures

Unit 5 - Mensuration

Chapter 23 - Perimeter And Area Of Plane Figures

23.1 Introduction

In mensuration, we deal with measurements of length, area, volume, surface area, etc. Knowledge of mensuration is of great use in our day-to-day life, specially, for intance, when we buy:

(i) cloth for shirts by length, (ii) a plot of land by area

(iii) milk, petrol, etc., by volume and so on.

23.2 Some Definitions

(a) (i) Perimeter:

The perimeter of a closed figure is the length of its boundary.

For example:

1. Perimeter of - ABC given alongside = Length of the boundary of - ABC = Length of AB + length of BC + length of CA = 3-5 cm + 4 cm + 4-5 cm = 12 cm

2. Perimeter of the plane figure (quadrilateral) ABCD given alongside = AB + BC + CD + DA = 5 cm + 7 cm + 6 cm + 8 cm = 26 cm

(ii) Unit of Perimeter:

The unit of perimeter is the same as the unit of length, i.e. centimetre (cm), metre (m), etc.

1. 1 cm = \(\frac{1}{100}\) m and 1 m = 100 cm

2. For finding the perimeter of any plane-figure convert each length into the same unit, e.g. if the lengths of the sides of a triangular figure are 80 cm, 1-2 m and 95 cm, its perimeter = 80 cm + 1-2 m + 95 cm = 80 cm + 120 cm + 95 cm = 295 cm

OR, perimeter of the given triangle = 80 cm + 1-2 m + 95 cm = 0-8 m + 1-2 m + 0-95 m = 2-95 m

(b) (i) Area:

The area of a plane figure is the measure of the size of the surface enclosed by its boundary.

For example:

The area of the given figure ABCDE is the measure of the size of shaded portion that is enclosed by its boundary.

(ii) Unit of Area:

If the unit of the length of each side of a plane figure is centimetre (cm), the unit of its area will be square-centimetre (sq. cm, i.e. cm²). In the same way, if the length of each side of a plane figure is metre (m), the unit of its area will be square-metre (sq. m, i.e. m²).

1 m = 100 cm and 1 m² = 100 × 100 cm² = 10,000 cm²

1 cm = \(\frac{1}{100}\) m and 1 cm² = \(\frac{1}{100}\) × \(\frac{1}{100}\) m² = \(\frac{1}{10,000}\) m²

Some other units in use:

For length: Millimetre (mm), Kilometre (km)

For area: Square millimetre (mm²), Square kilometre (km²)

23.3 Some Important Plane Figures

(a) Square:

A square is a four-sided closed figure with all sides equal and each angle 90°.

The figure given alongside shows a square ABCD in which AB = BC = CD = DA and \(\angle A = \angle B = \angle C = \angle D\) = 90°.

Let each side of the square be of length a units, i.e. AB = BC = CD = DA = a units.

The perimeter of the square = AB + BC + CD + DA = a + a + a + a = 4a = 4 × side of the square

and area of the square = its length × its breadth = a × a = a² = (side)².

1. Since the perimeter (P) of a square is given by the formula: P = 4 × length of its side

Length of each side of the square = \(\frac{\text{Its perimeter}}{4}\)

2. Since the area A of a square = (side)²

Length of its each side = \(\sqrt{A}\)

Example 1:

One side of a square is 6 cm. Find its perimeter and area.

Solution:

Perimeter of the square = 4 × side = 4 × 6 cm = 24 cm (Ans.)

Area of the square = (side)² = (6 cm)² = 36 cm² (Ans.)

Teacher's Note

Understanding perimeter and area helps students when calculating the amount of fencing needed for a garden or the cost of tiling a floor at home.

Example 2:

The perimeter of a square field is 96 m.

Find: (i) the length of its each side, (ii) the area of the square field.

Solution:

(i) Length of each side of the square = \(\frac{\text{its perimeter}}{4}\) = \(\frac{96}{4}\) m = 24 m (Ans.)

(ii) Area of the square field = (side)² = (24 m)² = 576 m² (Ans.)

Example 3:

The area of a square is 144 m².

Find: (i) its side (ii) its perimeter.

Solution:

(i) Side of the square = \(\sqrt{A}\) = \(\sqrt{144}\) m = \(\sqrt{2 \times 2 \times 2 \times 2 \times 3 \times 3}\) m = 2 × 2 × 3 m = 12 m (Ans.)

(ii) Perimeter of the square = 4 × side = 4 × 12 m = 48 m (Ans.)

Teacher's Note

Calculating square footage helps parents estimate paint quantities or flooring materials needed for home renovation projects.

Example 4:

Each side of a square field is 36 m. Find:

(i) its perimeter

(ii) its area

(iii) the cost of fencing the field at the rate of ₹ 20 per metre.

(iv) the cost of ploughing the field at the rate of ₹ 1-50 per m².

Length of fencing = Perimeter of the field.

Solution:

(i) Perimeter of the square field = 4 × its side = 4 × 36 m = 144 m (Ans.)

(ii) Area of the square field = (side)² = (36 m)² = 1,296 m² (Ans.)

(iii) Cost of fencing the field = Rate of fencing × perimeter = ₹ 20 × 144 = ₹ 2,880 (Ans.)

(iv) Cost of ploughing the field = Rate of ploughing × area = ₹ 1-50 × 1296 = ₹ 1,944 (Ans.)

Teacher's Note

Real-world applications like calculating fencing costs for a farm or ploughing expenses help students understand the practical value of geometric calculations in agriculture.

(b) Rectangle:

A rectangle is a four-sided closed figure of which the opposite sides are equal and each angle is 90°.

The adjacent figure shows a rectangle ABCD. Clearly, AB = CD = length (l) of the rectangle and, AD = BC = breadth (b) of the rectangle. Also, \(\angle A = \angle B = \angle C = \angle D\) = 90°.

Perimeter of rectangle ABCD = Length of its boundary = AB + BC + CD + DA = l + b + l + b = 2 (l + b) i.e. P = 2(l + b)

Opposite sides of a rectangle are equal

And area of rectangle = its length × its breath = l × b i.e. A = l + b

1. Since the perimeter of a rectangle is given by the formula P = 2 (l + b)

Its length, l = \(\frac{P}{2}\) - b and its breadth, b = \(\frac{P}{2}\) - l

2. Since the area of a rectangle is given by A = l × b

Its length, l = \(\frac{A}{b}\) and its breadth, b = \(\frac{A}{l}\)

Example 5:

The length and the breadth of a rectangle are 10 cm and 8 cm, respectively. Find its perimeter and area.

Solution:

Since the length of the rectangle (l) = 10 cm and the breadth of the rectangle (b) = 8 cm

Perimeter (P) = 2 (l + b) = 2 (10 + 8) cm = 36 cm (Ans.)

Area of rectangle (A) = l × b = 10 cm × 8 cm = 80 cm² (Ans.)

Teacher's Note

Knowing how to calculate rectangle dimensions helps when buying carpet for rooms or determining how much fabric is needed for a curtain.

Example 6:

The perimeter of a rectangle is 30 cm and its length is 8 cm. Find: (i) its breadth (ii) its area

Solution:

Given: P = 30 cm and l = 8 cm

(i) Breadth, b = \(\frac{P}{2}\) - l = \(\frac{30}{2}\) cm - 8 cm = (15 - 8) cm = 7 cm (Ans.)

and (ii) Area, A = l × b = 8 cm × 7 cm = 56 cm² (Ans.)

Example 7:

The area of a rectangular field is 450 m² and its width is 25 m. Find: (i) its length (ii) its perimeter (iii) the cost of fencing the field at the rate of ₹ 35-50 per metre.

Solution:

(i) Length = \(\frac{\text{Area}}{\text{Breadth}}\) = \(\frac{450}{25}\) m = 18 m (Ans.)

(ii) Perimeter = 2(l + b) = 2(18 + 25) m = 2 × 43 m = 86 m (Ans.)

(iii) Cost of fencing = Length of fence × Rate = 86 × ₹ 35-50 = ₹ 3,053 (Ans.)

Length of fence = Perimeter = 86 m

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