The scale of a map is 2 cm : 1 km.

1. Convert the actual distance to centimetres and express the scale using the same unit.

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

$2\text{ cm}:1\text{ km}=2\text{ cm}:100\,000\text{ cm}$

2. Rewrite the ratio in the form $1:n$.

$2:100\,000=1:50\,000$

1. Find the area represented by $4\text{ cm}^2$ on the map.

Since $2\text{ cm}$ on the map represents $1\text{ km}$ in actual distance,
$2\text{ cm} \times 2\text{ cm} \text{ on the map represents } 1\text{ km} \times 1\text{ km}.$
So,
$4\text{ cm}^2 \text{ on the map represents } 1\text{ km}^2.$

2. Use the area scale to convert $456\text{ cm}^2$ to the actual area.

$456\text{ cm}^2 \text{ on the map represents } \frac{456}{4}\text{ km}^2.$
$\frac{456}{4}=114$