ICSE Class 8 Maths Mensuration Chapter 32 Area and Perimeter of Plane Figures

Unit Six - Mensuration

Area and Perimeter of Plane Figures

Volume and Surface Area of Cuboids

Let's Recap

1. Find the area of a square, given its perimeter is 15.2 cm.

2. Find the perimeter of a 9.8 cm long rectangle, given its area is 53.9 cm².

3. Find the total surface area of a cube, given its volume is 216 cm³.

Are An are is a unit of measure for area equal to 100 square metres. The word, and the unit of measure, seems to have been created by the French and derived from the Latin word, area with its current meaning. The are is seldom used today, but its derivative form, the hectare is still a common unit of land measure in some countries.

Perimeter The word perimeter comes from the Greek word 'peri' (around) + 'metron' (measure).

Volume Volume is derived from the Latin word 'volvere', which means to turn or roll. The idea comes from the practice of writing on scrolls that were then rolled into a cylinder.

We'll have a 240 cm - 150 cm rectangular soft board hung on the wall, on which we can pin small charts. If the charts are of size 80 cm - 50 cm, then how many would we require to cover the entire board?

Thanks for your help Rahul! Well, the area of the rectangular board is 36000 cm² and the area of one chart is 4000 cm². That means, we can have nine such charts displayed on the board.

By Jove! That was quick, Priya

Why don't you answer that yourself?

Area and Perimeter of Plane Figures

Area - Perimeter

Circumference

Let us first recall the units of area and their conversions.

1 km² = 1 km - 1 km = 1000 m - 1000 m = 1000000 m²

1 hectare = 1 hm - 1 hm = 100 m - 100 m = 10000 m²

1 are = 1 dcm - 1 dcm = 10 m - 10 m = 100 m²

1 m² = 1 m - 1 m = 100 cm - 100 cm = 10000 cm²

1 m² = 1 m - 1 m = 1000 mm - 1000 mm = 1000000 mm²

1 cm² = 1 cm - 1 cm = 10 mm - 10 mm = 100 mm²

Area = length - breadth = 2835 m²

7x - 5x = 2835 m²

\[x^2 = \frac{2835}{35} \text{ m}^2 = 81 \text{ m}^2\]

\[x = \sqrt{81} \text{ m} = 9 \text{ m}\]

Thus, the length of the plot = 7 - 9 = 63 m and breadth of the plot = 5 - 9 = 45 m

Perimeter = 2(length + breadth)

= 2(63 + 45) m = 2 - 108 = 216 m

Cost of erecting fence along the boundary of the plot = 216 m - Rs 12.50 = Rs 2700.00

Area and Perimeter of Squares and Rectangles

Formulae

P = 4ℓ

(P = Perimeter of square, ℓ = length)

P = 2(ℓ + b)

(P = Perimeter of rectangle, ℓ = length, b = breadth)

A = ℓ²

(A = Area of square, ℓ = length)

A = ℓ - b

(A = Area of rectangle, ℓ = length, b = breadth)

Diagonal of a square = \(\sqrt{2 \text{ length}^2}\)

(by Pythagoras' theorem)

= \(\sqrt{2} \text{ length}\)

Diagonal of a rectangle = \(\sqrt{\text{length}^2 + \text{breadth}^2}\)

(by Pythagoras' theorem)

Example 1:

The area of a plot of land whose length is 7x and breadth is 5x is given as 2835 m². Find the cost of erecting a fence along its boundary at the rate of Rs 12.50 per metre.

Example 2:

The grass from a square lawn whose diagonal measures 6√2 m is transplanted to a rectangular lawn that is 4 m wide. What is the length of the rectangular lawn?

Diagonal of a square = length √2 = 6√2 m

length = \(\frac{6\sqrt{2}}{\sqrt{2}}\) = 6 m

Area of the square lawn = 6 - 6 = 36 m²

Area of rectangular lawn is also 36 m²

Width or breadth of the rectangular lawn = 4 m

As length - breadth = Area of rectangular lawn,

length - 4 m = 36 m²

length of the rectangular lawn = \(\frac{36}{4}\) m

= 9 m

Example 3:

The area of a square field is 9 hectares. A 3 m wide road is constructed along its boundary inside the field.

Teacher's Note

Understanding how to calculate the area and perimeter of simple shapes like squares and rectangles helps you determine how much material you need when building a garden bed or framing a picture.

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