A lake has an actual area of 2.56 km^2. It is represented by an area of 4 cm^2 on a map.

1. Convert the actual area to the same unit as the map area.

$1\text{ km}=100\,000\text{ cm}$

So
$1\text{ km}^2=(100\,000)^2\text{ cm}^2=10\,000\,000\,000\text{ cm}^2.$

Actual area:
$2.56\text{ km}^2=2.56\times 10\,000\,000\,000=25\,600\,000\,000\text{ cm}^2.$

2. Use the area ratio to find the linear scale.

Map area : actual area
$=4:25\,600\,000\,000$
$=1:6\,400\,000\,000.$

This is the area scale. If the linear scale is $1:n$, then the area scale is
$1:n^2.$

So
$n^2=6\,400\,000\,000$
$n=\sqrt{6\,400\,000\,000}=80\,000.$

Therefore the scale is
$1:80\,000.$

1. Apply the map scale to convert the map distance to actual distance.

From part (a), the scale is $1:8000$.

Actual distance
$=20\times 8000=160000\text{ cm}$

Convert to kilometres:
$160000\text{ cm}=\frac{160000}{100000}=1.6\text{ km}$