NCERT Class 8 Mathematics Ganita Prakash Part 2 Chapter 07 Area PDF Download

Chapter 7: Area

7.1 Rectangle and Squares

How many different ways can you divide a square into 4 parts of equal area?

One can actually think of infinitely many such ways! Consider a division, such as -

[Figure: A square divided into 4 equal parts in a 2×2 grid, See in your textbook]

and alter each part as follows.

[Figure: The same square with each part reshaped using compression and expansion along edges, See in your textbook]

In each part, the area is compressed along one edge and expanded along another edge. If both the compression and expansion are of the same magnitude, then all 4 parts still have the same area!

Math Talk

Try to think of different creative ways to divide a square into 4 parts of equal area.

You might have seen the rangoli art form, in which regions of different shapes are beautifully coloured using rangoli powder.

[Figure: A decorative rangoli design, See in your textbook]

Which of these rectangles requires more rangoli powder to be coloured, if the colouring is done evenly?

[Figure: Two rectangles side by side - one with dimensions 7 cm and 4 cm (pink), another with dimensions 8 cm and 3 cm (green), See in your textbook]

We can answer this by counting the number of non-overlapping unit squares (squares of sidelength 1 cm in this case) that can be packed into each of the rectangles.

Clearly, the rectangle having sidelengths 7 cm and 4 cm contains \( 7 \times 4 = 28 \) unit squares, and the rectangle having sidelengths 8 cm and 3 cm contains \( 8 \times 3 = 24 \) unit squares.

Thus, the rectangle of sidelengths 7 cm and 4 cm requires more powder to be coloured.

Recall that we measure the area of a region by finding the number of unit squares (which can also be a fraction) whose area equals that of the given region.

We have seen that the number of unit squares contained in a rectangle is given by the product of its length and width -

Area of a rectangle = length \( \times \) width.

The areas of the rectangles as seen in the previous problem are generally written as 28 sq. cm and 24 sq. cm, or as 28 cm² and 24 cm².

Teacher's Note

The formula length \( \times \) width works because you are literally counting how many unit squares fit along the length and how many fit along the width, then multiplying. If you have a 7 cm \( \times \) 4 cm rectangle, you get 7 unit squares in one direction and 4 in the other, giving \( 7 \times 4 = 28 \) squares total. Always multiply the two dimensions you are given.

What is the area of each triangle in this rectangle?

[Figure: A rectangle with dimensions 7 cm and 4 cm, with a diagonal line dividing it into two triangles, See in your textbook]

We have seen that the diagonal of a rectangle divides it into two congruent triangles. So, the area of each triangle is half the area of the rectangle.

In terms of unit squares, half the area fills exactly half the number of unit squares.

So the area of each triangle is \( \frac{1}{2} \times 7 \times 4 = 14 \) cm².

Why Can't Perimeter be a Measure of Area?

Why do we count the number of unit squares to assign measures for area? Couldn't we have just used the perimeter of a region, i.e., the length of its boundary as a measure of its area?

If two regions have the same perimeter, can't we conclude that they have the same area? Or, if one region has a larger perimeter than another region, can't we conclude that it also has a larger area?

The perimeter of a region is not indicative of its area. The reason is that regions can have the same perimeter but different areas, and vice versa. We can even find two regions, Region 1 and Region 2, such that

Perimeter of Region 1 \( > \) Perimeter of Region 2, but
Area of Region 1 \( < \) Area of Region 2.

Math Talk

Find two rectangles that are examples of such regions. If needed, use a grid paper (given at the end of the book) for this.

Also give an example of two regions of other shapes, where the region with the larger perimeter has the smaller area! This property should be visually clear in your example.

Teacher's Note

A common mistake is thinking that if a shape has a bigger perimeter, it must have a bigger area. This is false. For example, a very long thin rectangle can have a large perimeter but small area, while a more compact rectangle can have a smaller perimeter but larger area. When solving problems, always calculate area and perimeter separately - they measure different things.

Figure it Out

1. Identify the missing sidelengths.

[Figure (i): A composite shape made of rectangles with some dimensions and areas labeled. One rectangle has dimensions 7 in by 4 in with area 28 in². Another has area 35 in². Another has dimensions 3 in by ? in with area 21 in². Another has area 14 in². There is a missing dimension marked with ? in and a missing height marked with ? in., See in your textbook]

[Figure (ii): A composite rectangle divided into four parts. The top-left part is empty. The top-right parts have areas 29 m² and 11 m², with some dimensions marked as ? m. The bottom section has area marked as ?, and the total area is 50 m². One dimension is marked as 4 m., See in your textbook]

Key Points

• The area of a rectangle is found by multiplying its length and width. A unit square is a square with side length 1, and we count how many fit inside a shape.
• A diagonal line across a rectangle splits it into two equal triangles, so each triangle has area equal to half the rectangle's area.
• Perimeter (the distance around a shape) and area (the space inside a shape) are different measurements. Two shapes can have the same perimeter but different areas, or vice versa.

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