Area
We have studied the areas of closed figures (regular and irregular) in previous grades. Let us recall some key points.
The amount of region enclosed by a closed figure is called its
In previous grades, we arrived at the formula for the area of a rectangle and a square using square grid paper. Do you remember?
Note : Help students in recalling the method of finding the area of a rectangle and a square using grid papers. Provide square grid papers to students and let them come up with the formula.
Example 1
1. A floor is 5 m long and 4 m wide. A square carpet of sides 3 m is laid on the floor. Find the area of the floor that is not carpeted.
Solution
Length of the floor =
Width of the floor =
Area of the floor = length × width = 5 m × 4 m =
Length of the square carpet =
Area of the carpet = length × length = 3 m × 3 m =
Hence, the area of the floor laid with carpet is 9 sq m.
Therefore, the area of the floor that is not carpeted is: area of the floor minus the area of the floor laid with carpet = 20 sq m – 9 sq m =
Example 2
2.Four square flower beds each of side 4 m are in four corners on a piece of land 12 m long and 10 m wide. Find the area of the remaining part of the land.
Solution
Length of the land (l) =
Width of land (w) =
Area of the whole land = l × w = 12 m × 10 m =
The sidelength of each of the four square flower beds is (s) =
Area of one flower bed = s × s = 4 m × 4 m =
Hence, the area of the four flower beds = 4 × 16 sq m =
Therefore, the area of the remaining part of the land is: area of the complete land minus the area of all four flower beds = 120 sq m – 64 sq m =
Example 1
1. The area of a rectangular garden 25 m long is 300 sq m. What is the width of the garden?
Solution
Length of the rectangular garden =
Area of the garden =
The formula for area of rectangle is Area = length x width.
Substituting values: 300 =
Solve for width: width = 300 ÷ 25 =
Example 2
2. What is the cost of tiling a rectangular plot of land 500 m long and 200 m wide at the rate of ₹8 per hundred sq m?
Solution
Length of the rectangular plot =
Width of the rectangular plot =
Cost of tiling = Rs 8 per 100 m²
The formula for area of rectangle is Area = length x width.
Substituting the given values: 500 x 200 =
Now, the cost of tilling the rectangular plot per hundred sq m. = ₹8
So, the cost of tiling 1,00,000 sq m. of land =
Therefore, the cost of tiling 1,00,000 sq m. of land is ₹ 8000
Example 3
3. A rectangular coconut grove is 100 m long and 50 m wide. If each coconut tree requires 25 sq m, what is the maximum number of trees that can be planted in this grove?
Solution
Length of the rectangular grove =
Width of the rectangular grove =
Area required for each tree =
The formula for area of rectangle is Area = length x width.
Substituting the given values: 100 x 50 =
Since each tree requires 25 m², the number of trees that can be planted is:
Maximum trees = 5,000 ÷ 25 =
The grove can accommodate 200 coconut trees.
Example 4
- By splitting the following figures into rectangles, find their areas (all measures are given in metres)
Example 1
1. Explore and figure out how many pieces have the same area.
Solution
Example 2
2.How many times bigger is Shape D as compared to Shape C? What is the relationship between Shapes C, D and E?
Solution
Example 3
3. Which shape has more area: Shape D or F? Give reasons for your answer.
Solution
Example 4
1. Which shape has more area: Shape F or G? Give reasons for your answer.
Solution
Look at the figures below and guess which one of them has a larger area.
We can estimate the area of any simple closed shape by using a sheet of squared paper or graph paper where every square measures 1 unit × 1 unit or 1 square unit.
To estimate the area, we can trace the shape onto a piece of transparent paper and place the same on a piece of squared or graph paper and then follow the below conventions—
The area of one full small square of the squared or graph paper is taken as 1 sq unit.
Ignore portions of the area that are less than half a square.
If more than half of a square is in a region, just count it as 1 sq unit.
If exactly half the square is counted, take its area as
1 2
Find the area of the following figures.
Let’s Explore!
Why is area generally measured using squares?
Draw a circle on a graph sheet with diameter (breadth) of length 3. Count the squares and use them to estimate the area of the circular region.
As you can see, circles can’t be packed tightly without gaps in between. So, it is difficult to get an accurate measurement of area using circles as units.
Here, the same rectangle is packed in two different ways with circles—the first one has 42 circles and the second one has 44 circles.
Try using different shapes (triangle and rectangle) to fill the given space (without overlaps and gaps) and find out the merits associated with using a square shape to find the area rather than another shape. List out the points that make a square the best shape to use to measure area.
1. Find the area (in square metres) of the floor outside of the corridor.
2. Find the area (in square metres) occupied by your school playground.
Let’s Explore!
On a squared grid paper (1 square = 1 square unit), make as many rectangles as you can whose lengths and widths are a whole number of units such that the area of the rectangle is 24 square units.
a. Which rectangle has the greatest perimeter?
b. Which rectangle has the least perimeter?
If you take a rectangle of area 32 sq cm, what will your answers be?
Given any area, is it possible to predict the shape of the rectangle with the greatest perimeter as well as the least perimeter? Give examples and reasons for your answer.