A circular garden has radius 21 metres. A uniform path of width 3.5 metres is constructed all around the outside of the garden. What is the area, in square metres, of just the path?

Introduction / Context:
This is a standard question on areas of concentric circles. A circular garden has a path of uniform width built around it, and we are asked to find the area of the path only. This type of problem is frequently tested in aptitude and entrance exams and reinforces understanding of the area formula for circles and the effect of adding a uniform border or path.

Given Data / Assumptions:

• The radius of the circular garden (inner circle) is 21 m.
• A path of uniform width 3.5 m is constructed outside the garden.
• The path forms a ring between the inner circle and a larger outer circle.
• We must find the area of this ring (the path) in square metres.
• Use π = 22/7, which is consistent with typical exam settings and yields neat fractional answers.

Concept / Approach:
The area of a circle of radius r is given by: Area = π * r^2 Here, the path area is the difference between the area of the outer circle (garden plus path) and the area of the inner circle (garden alone). If r is the inner radius and R is the outer radius, then: Area of path = π * R^2 − π * r^2 = π * (R^2 − r^2) We know r = 21 m and the path width is 3.5 m, so R = 21 + 3.5 = 24.5 m.

Step-by-Step Solution:
Step 1: Write the known values. Inner radius r = 21 m. Path width = 3.5 m, so outer radius R = 21 + 3.5 = 24.5 m. Step 2: Use the ring area formula. Area of path = π * (R^2 − r^2). Step 3: Compute R^2 and r^2. R^2 = 24.5^2 = 600.25 r^2 = 21^2 = 441 R^2 − r^2 = 600.25 − 441 = 159.25 Step 4: Use π = 22/7. Area of path = (22 / 7) * 159.25 Multiply 159.25 by 22: 159.25 * 22 = 3503.5 Now divide by 7: 3503.5 / 7 = 500.5 square metres.

Verification / Alternative check:
We can also compute the areas separately. Inner circle: Area = (22 / 7) * 21^2 = (22 / 7) * 441 = 22 * 63 = 1386 sq m. Outer circle: Area = (22 / 7) * 24.5^2 = (22 / 7) * 600.25 = 22 * (600.25 / 7) = 22 * 85.75 = 1886.5 sq m. The difference 1886.5 − 1386 = 500.5 sq m matches the ring area calculated earlier, confirming that 500.5 sq m is correct.

Why Other Options Are Wrong:
Option 2: 505.0 sq m is slightly higher than the correct value, which could result from rounding or miscalculating one of the squares. Option 3: 452.4 sq m is significantly lower and does not match the area difference between the two circles. Option 4: 550.0 sq m would require a larger difference in squared radii or a different value of π; it does not come from the given data. Option 5: None of these is incorrect because the exact value 500.5 sq m is present among the options.

Common Pitfalls:
Students sometimes treat 3.5 m as the new radius instead of the extra width or forget to add 3.5 m to obtain the outer radius. Another common mistake is computing only the area of the larger circle and forgetting to subtract the inner circle area. Algebraic errors when squaring 24.5 or simple arithmetic mistakes when subtracting 441 can also lead to incorrect answers. Carefully writing each step and using the difference of squares approach helps avoid these problems.

Final Answer:
The area of the path is 500.5 sq m.