NCERT Solutions for Class 7 Maths Exercise 11.4: Perimeter and Area

Visualizing Nested Shapes: Paths and Borders

To solve problems in Exercise 11.4, you need to visualize nested rectangles. Imagine a rectangular school playground. If a running track of uniform width is built around it on the outside, we get a new, larger rectangle. The area of this track is simply the difference between the outer rectangle and the inner playground rectangle. Conversely, if a path is built along the inner boundary of a park, the path's area is the difference between the original park's area and the remaining central rectangular space. The key is adjusting the dimensions: if a path of width 'w' is added outside a rectangle of length 'L' and breadth 'B', the new dimensions become (L + 2w) and (B + 2w) because the width is added on both left-right and top-bottom sides. Subtracting the original area from the new area gives you the exact area of the path.

Step-by-Step Method: Calculating the Area of Cross-Roads

1. Identify Individual Road Dimensions — Find the dimensions of the two perpendicular roads. One road runs parallel to the length of the rectangular field (Area = Length Road Width), and the other runs parallel to the breadth (Area = Breadth Road Width).
2. Calculate the Overlapping Center — Notice where the two roads intersect in the middle. This common region is a square or rectangle with sides equal to the width of the roads. Calculate its area (Area of Center = Road Width * Road Width).
3. Subtract the Overlap to Avoid Double Counting — If you simply add the areas of the two roads, you will count the intersection twice. To get the correct total road area, use the formula: Total Road Area = (Area of Road 1 + Area of Road 2) - Overlap Area.
4. Convert Units if Required — If the question asks for the answer in hectares, convert square meters to hectares. Note that 1 Hectare = 10,000 square meters.

Exam Tips & Common Unit Conversions

Many students lose marks because they forget to convert units properly. Always remember:

1. Linear vs Square Conversions: Since 1 meter = 100 centimeters, then 1 square meter (1 m²) is equal to 100 cm * 100 cm = 10,000 cm².
2. Hectare Conversions: 1 hectare = 10,000 m². To convert m² into hectares, divide the value by 10,000.
3. Double Counting Mistake: In cross-road problems, never forget to subtract the square area of the intersection! Leaving out this subtraction step is the most common mistake in class tests.

Practice Questions with Solutions

• Q: A rectangular park is 45 m long and 30 m wide. A path 2.5 m wide is constructed outside the park. Find the area of the path. A: Step 1: Write down the dimensions of the inner park. Length (l) = 45 m Breadth (b) = 30 m Area of the inner park = l b = 45 30 = 1350 m² Step 2: Find the dimensions of the outer rectangle. Since a 2.5 m wide path is built outside, we add 2.5 m on both sides. Outer Length (L) = 45 + 2(2.5) = 45 + 5 = 50 m Outer Breadth (B) = 30 + 2(2.5) = 30 + 5 = 35 m Area of the outer rectangle = L B = 50 35 = 1750 m² Step 3: Calculate the area of the path. Area of path = Outer Area - Inner Area = 1750 - 1350 = 400 m² Final answer: The area of the path is 400 m².
• Q: A path 5 m wide runs along the inside of a square park of side 100 m. Find the area of the path. Also find the cost of cementing it at the rate of Rs. 250 per 10 m². A: Step 1: Find the area of the outer square park. Side of the outer square (S) = 100 m Area of the outer square = S² = 100 100 = 10,000 m² Step 2: Find the dimensions and area of the inner square. Since the 5 m path runs inside, we subtract 5 m from both sides. Side of the inner square (s) = 100 - 2(5) = 100 - 10 = 90 m Area of the inner square = s² = 90 90 = 8,100 m² Step 3: Calculate the area of the path. Area of path = Outer Area - Inner Area = 10,000 - 8,100 = 1,900 m² Step 4: Find the cost of cementing. Rate = Rs. 250 for 10 m², which means Rs. 25 per 1 m². Total Cost = Area Rate = 1900 (250 / 10) = 1900 * 25 = Rs. 47,500 Final answer: The area of the path is 1,900 m² and the cost of cementing is Rs. 47,500.
• Q: Two cross roads, each of width 10 m, cut at right angles through the centre of a rectangular park of length 700 m and breadth 300 m and parallel to its sides. Find the area of the roads in hectares. A: Step 1: Find the area of the road parallel to length. Length of road = 700 m, Width = 10 m Area of Road 1 = 700 10 = 7,000 m² Step 2: Find the area of the road parallel to breadth. Breadth of road = 300 m, Width = 10 m Area of Road 2 = 300 10 = 3,000 m² Step 3: Find the area of the common overlapping intersection. Intersection is a square of side 10 m. Area of overlap = 10 * 10 = 100 m² Step 4: Calculate total area of roads. Total Road Area = Area of Road 1 + Area of Road 2 - Overlap Area Total Road Area = 7,000 + 3,000 - 100 = 9,900 m² Step 5: Convert the area into hectares. 1 hectare = 10,000 m² Area in hectares = 9,900 / 10,000 = 0.99 hectares Final answer: The area of the cross roads is 9,900 m² or 0.99 hectares.
• Q: Convert: (a) 5.8 hectares to m² (b) 2.5 m² to cm². A: Step 1: For part (a), we know 1 hectare = 10,000 m². 5.8 hectares = 5.8 10,000 = 58,000 m² Step 2: For part (b), we know 1 m² = 10,000 cm². 2.5 m² = 2.5 10,000 = 25,000 cm² Final answer: (a) 58,000 m², (b) 25,000 cm².

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