RBSE Solutions Class 6 Maths Chapter 14 Perimeter and Area Exercise 14.2

Question 1. By counting squares, estimate the areas of the figures

Answer: To find the area of the figures, we count the number of squares they cover. We count a full square as 1 unit, a square that is more than half-filled as 1 unit, and a square that is half-filled as 0.5 units. Squares that are less than half-filled are ignored. The table above shows the estimated area for each figure by applying these rules. Counting squares helps estimate areas of irregular shapes when precise measurements are not available.
In simple words: We find the area by counting how many small squares each shape covers. Full squares and more-than-half squares count as one, half squares count as half, and small parts are ignored.

🎯 Exam Tip: When estimating areas by counting squares, be careful to distinguish between full, more than half, and half-filled squares to get the most accurate estimate.

Question 2. Find the areas of the following figures. What do you infer from this?
Answer:
For the first figure (a square):
Side = 4 cm
Area = \( (\text{side})^2 = (4)^2 = 16 \) sq. cm

For the second figure (a rectangle):
Length = 8 cm
Breadth = 2 cm
Area = \( \text{Length} \times \text{Breadth} = 8 \times 2 = 16 \) sq. cm

For the third figure (a rectangle):
Length = 16 cm
Breadth = 1 cm
Area = \( \text{Length} \times \text{Breadth} = 16 \times 1 = 16 \) sq. cm

From these calculations, we can infer that all three figures have the same area (16 sq. cm) even though they have different shapes and dimensions. This shows that different shapes can enclose the same amount of space.
In simple words: We found that all three shapes, even though they look different, have the same area of 16 square centimeters. This means different shapes can cover the same amount of space.

🎯 Exam Tip: Remember that area measures the space inside a shape. Shapes can look very different but still have the same area if they enclose the same amount of surface.

Question 4. A room is 10 m long and 8 m wide. How many square meters of carpet is required to cover the floor of the room?
Answer:
To cover the floor of the room, the amount of carpet needed will be equal to the area of the floor.
Length of the room (l) = 10 m
Width of the room (b) = 8 m
Area of the floor = \( \text{l} \times \text{b} = 10 \times 8 = 80 \) sq. m
Therefore, 80 square meters of carpet will be required to cover the floor of the room. The area helps in calculating the material needed.
In simple words: The room is 10 meters long and 8 meters wide. To find how much carpet is needed, we multiply the length by the width. This gives us 80 square meters of carpet.

🎯 Exam Tip: Always make sure the units are the same (e.g., all in meters or all in centimeters) before calculating the area.

Question 6. Find the areas of the following rectangles. Which rectangle has least area and which one has greatest area ?
(i) l = 2 m, b = 80 cm
(ii) l = 180 cm, b = 70 cm
(iii) l = 200 cm, b = 1 m
(iv) l = 190 cm, b = 1 m
Answer:
First, we will calculate the area for each rectangle, converting all measurements to centimeters for consistency.

(i) Length (l) = 2 m = \( 2 \times 100 \) cm = 200 cm
Breadth (b) = 80 cm
Area = \( \text{l} \times \text{b} = 200 \times 80 = 16000 \) sq. cm

(ii) Length (l) = 180 cm
Breadth (b) = 70 cm
Area = \( \text{l} \times \text{b} = 180 \times 70 = 12600 \) sq. cm

(iii) Length (l) = 200 cm
Breadth (b) = 1 m = \( 1 \times 100 \) cm = 100 cm
Area = \( \text{l} \times \text{b} = 200 \times 100 = 20000 \) sq. cm

(iv) Length (l) = 190 cm
Breadth (b) = 1 m = \( 1 \times 100 \) cm = 100 cm
Area = \( \text{l} \times \text{b} = 190 \times 100 = 19000 \) sq. cm

Comparing the areas:
20000 > 19000 > 16000 > 12600

Therefore, rectangle (iii) has the greatest area (20000 sq. cm) and rectangle (ii) has the least area (12600 sq. cm). This comparison helps in understanding relative sizes.
In simple words: We found the area for each rectangle. Rectangle (iii) has the biggest area, and rectangle (ii) has the smallest area. We had to change meters to centimeters first to compare them fairly.

🎯 Exam Tip: Always convert all units to be the same (e.g., all centimeters or all meters) before performing calculations to avoid errors.

Question 8. Six square flower beds each of side 1 m are dug on a piece of land 8 m long 6 m wide. What is the area of the remaining part of the land?
Answer:
First, let's find the total area of the land:
Length of land = 8 m
Width of land = 6 m
Area of land = \( \text{Length} \times \text{Width} = 8 \times 6 = 48 \) sq. m

Next, let's find the area of one square flower bed:
Side of each flower bed = 1 m
Area of one flower bed = \( (\text{side})^2 = 1 \times 1 = 1 \) sq. m

Since there are six square flower beds, their total area is:
Total area of flower beds = \( \text{Area of one bed} \times \text{Number of beds} = 1 \times 6 = 6 \) sq. m

Finally, to find the area of the remaining part of the land, subtract the total area of the flower beds from the total area of the land:
Remaining area = \( 48 - 6 = 42 \) sq. m
Thus, the area of the remaining land will be 42 sq. m. Calculating individual areas and subtracting helps in solving problems involving composite shapes.
In simple words: The whole land is 48 square meters. Six flower beds, each 1 square meter, take up 6 square meters. So, the land left over is 48 minus 6, which is 42 square meters.

🎯 Exam Tip: When dealing with areas that have parts removed, always calculate the total area first, then the area of the removed parts, and finally subtract to find the remaining area.

Question 9. What will be the change in the area of a rectangle if its
(i) Length and breadth are both doubled?
(ii) Length is tripled and breadth is doubled twice?
Answer:
(i) Let the initial length of the rectangle be 'a' and the initial breadth be 'b'.
Initial area = \( \text{a} \times \text{b} = \text{ab} \)
If the length and breadth are both doubled:
New length = \( 2\text{a} \)
New breadth = \( 2\text{b} \)
New area = \( (2\text{a}) \times (2\text{b}) = 4\text{ab} \)
Since the initial area was 'ab', the new area is \( 4 \times (\text{initial area}) \).
Thus, the area of the rectangle becomes 4 times its initial area. Doubling both dimensions makes the area four times larger.

(ii) Let the initial length of the rectangle be 'a' and the initial breadth be 'b'.
Initial area = \( \text{a} \times \text{b} = \text{ab} \)
If the length is tripled and the breadth is doubled:
New length = \( 3\text{a} \)
New breadth = \( 2\text{b} \)
New area = \( (3\text{a}) \times (2\text{b}) = 6\text{ab} \)
Since the initial area was 'ab', the new area is \( 6 \times (\text{initial area}) \).
Thus, the area of the rectangle becomes 6 times its initial area. Multiplying dimensions by different factors also multiplies the area by the product of those factors.
In simple words: (i) If you double both the length and width of a rectangle, its new area will be 4 times bigger. (ii) If you triple the length and double the width, the new area will be 6 times bigger.

🎯 Exam Tip: Remember that if you multiply the length by 'x' and the breadth by 'y', the area will be multiplied by 'x * y'.

Question 10. What will be the change in the area of a square if its side is
(i) Halved?
(ii) Doubled?
Answer:
(i) Let the initial side of the square be 'a'.
Initial area = \( \text{a} \times \text{a} = \text{a}^2 \)
If the side is halved:
New side = \( \frac{1}{2}\text{a} \)
New area = \( (\frac{1}{2}\text{a}) \times (\frac{1}{2}\text{a}) = \frac{1}{4}\text{a}^2 \)
Since the initial area was \( \text{a}^2 \), the new area is \( \frac{1}{4} \times (\text{initial area}) \).
Thus, the new area remains one-fourth of the initial area. Halving the side makes the area four times smaller.

(ii) Let the initial side of the square be 'a'.
Initial area = \( \text{a} \times \text{a} = \text{a}^2 \)
If the side is doubled:
New side = \( 2\text{a} \)
New area = \( (2\text{a}) \times (2\text{a}) = 4\text{a}^2 \)
Since the initial area was \( \text{a}^2 \), the new area is \( 4 \times (\text{initial area}) \).
Thus, the new area becomes four times the initial area. Doubling the side makes the area four times larger.
In simple words: (i) If you cut the side of a square in half, its area becomes one-fourth of what it was before. (ii) If you double the side of a square, its area becomes four times bigger.

🎯 Exam Tip: For a square, the area changes by the square of the factor by which its side is changed. If the side is multiplied by 'k', the area is multiplied by \( \text{k}^2 \).