The scale of a map is 8 cm : 2 km.

1. Convert the actual distance to centimetres and simplify the ratio to the form $1:n$.

$2\text{ km}=2000\text{ m}=200000\text{ cm}$

So the scale is
$8\text{ cm}:200000\text{ cm}=8:200000.$

Divide both sides by $8$:
$8:200000=1:25000.$

1. Convert the actual area from square metres to square centimetres.

Since $1\text{ m}=100\text{ cm}$,

$1\text{ m}^2=100^2\text{ cm}^2=10\,000\text{ cm}^2$

$90\,000\text{ m}^2=90\,000\times 10\,000=900\,000\,000\text{ cm}^2$

2. Apply the area scale factor to find the lake area on the map.

From part (a), the scale is $1:25\,000$.

So the area scale is

$1^2:25\,000^2$

Map area

$=\frac{900\,000\,000}{25\,000^2}$

$=1.44\text{ cm}^2$