Understanding the Problem: Rectangular Garden and Path Area

The problem asks us to calculate the area of a path built around a rectangular garden. We are given the dimensions of the garden and the uniform width of the path surrounding it.

Given Information:

• Garden Length: 220 metres
• Garden Width: 70 metres
• Path Width: 4 metres

The path is built around the garden. This means the path forms a larger rectangle outside the garden. To find the area of the path, we need to find the area of the larger rectangle (garden plus path) and subtract the area of the inner rectangle (the garden itself).

Step-by-Step Calculation of the Path Area

Step 1: Calculate the Area of the Garden

The garden is a rectangle. The area of a rectangle is given by the formula: Area = Length × Width.

Area of Garden = Garden Length × Garden Width

\(\text{Area}_{\text{garden}} = 220 \text{ m} \times 70 \text{ m}\)

\(\text{Area}_{\text{garden}} = 15400 \text{ m}^2\)

Step 2: Calculate the Dimensions of the Garden Including the Path

Since the path of width 4 metres is built all around the garden, it adds 4 metres to each end of both the length and the width.

• New Length (including path) = Garden Length + 2 × Path Width
• New Width (including path) = Garden Width + 2 × Path Width

New Length = 220 m + 2 \(\times\) 4 m

New Length = 220 m + 8 m

New Length = 228 m

New Width = 70 m + 2 \(\times\) 4 m

New Width = 70 m + 8 m

New Width = 78 m

Step 3: Calculate the Area of the Garden Including the Path

This larger area is also a rectangle with the new calculated dimensions.

Area (Garden + Path) = New Length × New Width

\(\text{Area}_{\text{total}} = 228 \text{ m} \times 78 \text{ m}\)

To calculate \(228 \times 78\):

Summing the values: \(14000 + 1400 + 560 + 1600 + 160 + 64 = 17784\)

\(\text{Area}_{\text{total}} = 17784 \text{ m}^2\)

Step 4: Calculate the Area of the Path

The area of the path is the difference between the total area (garden plus path) and the area of the garden alone.

Area of Path = Area (Garden + Path) - Area of Garden

\(\text{Area}_{\text{path}} = \text{Area}_{\text{total}} - \text{Area}_{\text{garden}}\)

\(\text{Area}_{\text{path}} = 17784 \text{ m}^2 - 15400 \text{ m}^2\)

\(\text{Area}_{\text{path}} = 2384 \text{ m}^2\)

Conclusion

The area of the path built around the rectangular garden is 2384 square metres.

Revision Table: Rectangular Area Calculations

Additional Information: Areas of Rectangular Shapes

Calculating areas involving paths around rectangular shapes is a common geometry problem. The key idea is to consider the shapes as nested rectangles. The area of the path is the area of the outer rectangle minus the area of the inner rectangle.

• When a path is *outside* a rectangle, both the length and width of the outer rectangle are increased by twice the path width.
• When a path is *inside* a rectangle, both the length and width of the inner rectangle are decreased by twice the path width. The area of the path would then be the area of the outer rectangle minus the area of the inner rectangle.

Understanding how the path width affects the overall dimensions is crucial for accurately calculating the areas.