A circular path of 3 m width runs around a circular park with radius 8 m. What is the area of the path?

Correct — A.
The path runs outside the park, so:
inner radius r = 8 m, outer radius R = 8 + 3 = 11 m

Path area = π(R² − r²) = π(121 − 64)
= 57π m²

With π ≈ 3.1416: 57 × 3.1416 ≈ 179.07 m² → option (a)

Why the others are wrong

• (b)
  131.09 m² — 131.09 m² is below 150.80 m², which is 3 m × the park's own edge (2π × 8). All of the path lies outside that edge, so its area must be larger than 150.80.
• (c)
  138.02 m² — 138.02 m² is about 43.9π, but the ring is π(11² − 8²) = 57π. It also falls below the 150.80 m² floor set by the park's edge × 3 m.
• (d)
  144.05 m² — 144.05 m² is the nearest of the three to the 150.80 m² floor but still under it, about 35 m² short of the true 179.07.

A path around a circle is a ring (an annulus): the big circle minus the park. Its area is π(R² − r²), with R the outer radius and r the inner.

A second route gives the same number: area = width × circumference at the middle of the path. The middle radius is 9.5 m, so 3 × 2π × 9.5 = 57π, because R² − r² = (R + r)(R − r) = 19 × 3.

Key facts

• Ring area = π(R² − r²) = π(R + r)(R − r).
• A path of width w around a circle of radius r has outer radius r + w.
• Here the path is 57π ≈ 179.07 m², and the park inside it is 64π ≈ 201.06 m².

Common traps

• Adding half the width (8 + 1.5) as if the 3 m were shared between two sides: the full width adds to the radius.
• Computing π × 11² and forgetting to subtract the park, which gives about 380.13 m².

No directly related past PYQ was found.