A square lawn is surrounded by a path 2.5 m wide. If area of the path is 165 m², find the area of the lawn.

Let ABCD be the square lawn of side ‘x’ m that is AB = x.

Given path is 2.5 m wide (shaded region).

So, side of square EFGH (lawn along with the path) as shown in the above figure is given by

EF = (x + 2.5 + 2.5) m

⇒ EF = (x + 5) m

We have area of lawn equal to the difference between the area of the lawn along with the path and the area of the path (shaded region).

⇒ area of ABCD = area of EFGH – area of the shaded region

We know, Area of a square = side²

So, we have Area of square ABCD = AB²

∴ Area of ABCD = x² m²

Similarly, we have Area of square EFGH = EF²

∴ Area of EFGH = (x + 5)² m²

Given area of the path = 165 m²

⇒ x² = (x + 5)² – 165

⇒ (x + 5)² – x² = 165

⇒ x² + 10x + 25 – x² = 165

⇒ 10x + 25 = 165

⇒ 10x = 140

So, side of the square lawn is 14 m.

∴ Area of the square lawn = 14² = 196 m²

Thus, area of the lawn is 196 m².