  The area-perimeter scaling can be employed to evaluate the fractal dimension
of urban boundaries. However, the formula in common use seems to be not
correct. By means of mathematical method, a new formula of calculating the
boundary dimension of cities is derived from the idea of box-counting
measurement and the principle of dimensional consistency in this paper. Thus,
several practical results are obtained as follows. First, I derive the
hyperbolic relation between the boundary dimension and form dimension of
cities. Using the relation, we can estimate the form dimension through the
boundary dimension and vice versa. Second, I derive the proper scales of
fractal dimension: the form dimension comes between 1.5 and 2, and the boundary
dimension comes between 1 and 1.5. Third, I derive three form dimension values
with special geometric meanings. The first is 4/3, the second is 3/2, and the
third is about 1.7071. The fractal dimension relation formulae are applied to
China's cities and the cities of the United Kingdom, and the computations are
consistent with the theoretical expectation. The formulae are useful in the
fractal dimension estimation of urban form, and the findings about the fractal
parameters are revealing for future city planning and the spatial optimization
of cities.
