Define the relationship

The relationship between area scale and linear scale is defined by the square of the linear scale ratio ($L^2 = A$).

$\text{Scale}^2 = \frac{Area_{map}}{Area_{ground}}$

Substitute the given values

Substitute the given areas into the formula to find the square of the scale ratio.

$\text{Scale}^2 = \frac{100 \text{ cm}^2}{10000 \text{ m}^2} = \frac{1}{100} \frac{\text{cm}^2}{\text{m}^2}$

Solve for the linear scale

Take the square root of both sides to find the linear relationship.

$\text{Scale} = \sqrt{\frac{1}{100}} = \frac{1}{10} \frac{\text{cm}}{\text{m}}$

Final conversion

The result $\frac{1}{10} \frac{\text{cm}}{\text{m}}$ implies that 1 cm on the map represents 10 m on the ground.

$1 \text{ cm} = 10 \text{ m}$