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Perimeter and Area - Area of Pathways or Borders

Grade 7CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Definition of a Path or Border: A path is a region of uniform width that surrounds a rectangular or square area, either on the inside or the outside. Visually, this creates a 'frame' effect where two rectangles (outer and inner) share the same center point.

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Outer Path Construction: When a path is built outside a rectangle, the dimensions of the larger outer rectangle are found by adding twice the width of the path to the original length and breadth. Visually, the path extends the boundaries of the original shape in all four directions.

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Inner Path Construction: When a path is built inside a rectangle (like a border in a park), the dimensions of the inner area are found by subtracting twice the width of the path from the outer dimensions. Visually, the inner area appears as a smaller rectangle 'floating' inside the boundary of the larger one.

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Calculating Path Area: The area of a path is the difference between the area of the outer rectangle and the area of the inner rectangle. This is represented visually as the shaded 'ring' or 'border' between the two rectangular boundaries.

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Crossroads within a Rectangle: These are two perpendicular paths (one horizontal and one vertical) that run through the center of a rectangular field. Visually, they form a '+' shape. A crucial point is that the small square where the two paths intersect is counted in both paths, so its area must be subtracted once to get the correct total path area.

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Unit Consistency: All measurements, including the length, breadth, and the width of the path, must be in the same units (e.g., all meters or all centimeters) before performing calculations. If costs for tiling or gravelling are given, ensure the area units match the rate units (e.g., m2m^2 for rate per square meter).

📐Formulae

Area of a Rectangle=length×breadth\text{Area of a Rectangle} = \text{length} \times \text{breadth}

Area of a Square=side×side\text{Area of a Square} = \text{side} \times \text{side}

Outer Length (L)=l+2w\text{Outer Length } (L) = l + 2w (where ll is inner length and ww is path width)

Outer Breadth (B)=b+2w\text{Outer Breadth } (B) = b + 2w (where bb is inner breadth)

Inner Length (l)=L−2w\text{Inner Length } (l) = L - 2w (where LL is outer length)

Inner Breadth (b)=B−2w\text{Inner Breadth } (b) = B - 2w (where BB is outer breadth)

Area of Path=(L×B)−(l×b)\text{Area of Path} = (L \times B) - (l \times b)

Area of Crossroads=(Area of Path 1+Area of Path 2)−(width of path)2\text{Area of Crossroads} = (\text{Area of Path 1} + \text{Area of Path 2}) - (\text{width of path})^2

💡Examples

Problem 1:

A rectangular park is 45m45 m long and 30m30 m wide. A path 2.5m2.5 m wide is constructed outside the park. Find the area of the path.

Solution:

  1. Identify inner dimensions: Inner length l=45ml = 45 m, Inner breadth b=30mb = 30 m. \n 2. Calculate inner area: Inner Area=45m×30m=1350m2\text{Inner Area} = 45 m \times 30 m = 1350 m^2. \n 3. Identify path width: w=2.5mw = 2.5 m. \n 4. Calculate outer dimensions: \n Outer length L=45+(2×2.5)=45+5=50mL = 45 + (2 \times 2.5) = 45 + 5 = 50 m. \n Outer breadth B=30+(2×2.5)=30+5=35mB = 30 + (2 \times 2.5) = 30 + 5 = 35 m. \n 5. Calculate outer area: Outer Area=50m×35m=1750m2\text{Outer Area} = 50 m \times 35 m = 1750 m^2. \n 6. Calculate area of path: Area of Path=Outer Area−Inner Area=1750m2−1350m2=400m2\text{Area of Path} = \text{Outer Area} - \text{Inner Area} = 1750 m^2 - 1350 m^2 = 400 m^2.

Explanation:

Since the path is outside, we add twice the width to both the length and breadth to find the larger rectangle's dimensions. The path's area is the difference between the large and small rectangle areas.

Problem 2:

Two crossroads, each of width 5m5 m, run at right angles through the center of a rectangular park of length 70m70 m and breadth 45m45 m and parallel to its sides. Find the area of the roads.

Solution:

  1. Identify Path 1 (parallel to length): Dimensions are 70m×5m70 m \times 5 m. \n Area of Path 1=70×5=350m2\text{Area of Path 1} = 70 \times 5 = 350 m^2. \n 2. Identify Path 2 (parallel to breadth): Dimensions are 45m×5m45 m \times 5 m. \n Area of Path 2=45×5=225m2\text{Area of Path 2} = 45 \times 5 = 225 m^2. \n 3. Identify the common intersection: It is a square with side 5m5 m. \n Area of Common Square=5×5=25m2\text{Area of Common Square} = 5 \times 5 = 25 m^2. \n 4. Calculate total area of roads: Total Area=(Area of Path 1+Area of Path 2)−Common Area\text{Total Area} = (\text{Area of Path 1} + \text{Area of Path 2}) - \text{Common Area}. \n Total Area=(350+225)−25=575−25=550m2\text{Total Area} = (350 + 225) - 25 = 575 - 25 = 550 m^2.

Explanation:

The paths overlap at the center. If we simply add the areas of the two rectangular strips, the middle 5m×5m5 m \times 5 m square is counted twice. We subtract it once to get the actual area covered by the roads.