A circular garden having a diameter of 10 m is surrounded by a concrete path that is 1 m wide. Calculate the percentage increase in area resulting from the addition of the path.

Correct — C. A 1 m path on every side adds 1 m to the radius.

Garden radius: 10 ÷ 2 = 5 m
Outer radius, garden plus path: 5 + 1 = 6 m

Added area: π(6² − 5²) = π(36 − 25) = 11π m²
Percentage increase: 11π ÷ 25π × 100 = 44% → option (c)

Why the others are wrong

• (a)
  18% — 18% is less than the 20% rise in the radius itself, and area grows faster than the radius: 1.2² = 1.44, a 44% rise.
• (b)
  20% — 20% is the rise in the radius, 5 m to 6 m. Area goes with the square of the radius, so it rises by 1.2² − 1 = 0.44, that is 44%.
• (d)
  54% — 54% overshoots: it would make the new area 1.54 × 25π = 38.5π m², but a 6 m radius gives exactly 36π m².

Area grows with the square of the radius. A path of uniform width w around a circle of radius r makes the outer radius r + w, and the total area π(r + w)².

Here the radius rises 20%, from 5 m to 6 m, so the area multiplies by 1.2² = 1.44, a 44% increase. Working with the ratio means π never has to be multiplied out.

Key facts

• Area of a circle = πr², so multiplying the radius by k multiplies the area by k².
• Area of a circular path = π(R² − r²) = π(R + r)(R − r), here π × 11 × 1 = 11π m².
• A path of width w on every side adds w to the radius and 2w to the diameter.

Common traps

• Adding the path width to the diameter only once: a 1 m path on every side makes the diameter 12 m, not 11 m.
• Using the 10 m diameter as the radius: the areas then compare as 11² : 10², a 21% rise, not 44%.

No directly related past PYQ was found.