Theoretical Relationships Between Building Setback, Plot Frontage, and Plot Depth in Terms of Road and Building Densities

A theoretical investigation into the relationships between building setback, plot frontage, and plot depth in terms of building density (number of buildings per unit area) and road-network density (total length of road networks per unit area) is sparsely documented. Hence, in this study, we constructed a mathematical model to analyze the above relationship and maximize the average plot frontage. We found that (1) the average plot frontage is inversely proportional to the gross building density and is a convex upward quadratic function of the road-network density; (2) the average plot depth and setback allowance do not depend on the building density and are a monotonically decreasing function of the road-network density; and (3) the ratio of average plot depth to frontage is proportional to the gross building density and is a monotonically decreasing function of the road-network density. Based on the above findings, we developed a criterion for determining whether the average plot frontage is longer than the average plot depth. Further, the model was validated via an empirical analysis of the Tokyo metropolitan region. The results revealed that the modes of the average plot frontage, average plot depth, and ratio of plot depth to frontage were 11 m, 16 m, and 1.4, respectively. The mode of setback allowance was 5 m. The ratio of building setback to the setback allowance was in the range 0.4–0.6, which indicated that typically approximately half of the allowance is assigned to building setback. The findings of this study will provide urban planners and practitioners with a firm theoretical foundation for understanding the relationship between building setbacks, plot shape, building density, and road-network density in the district scale.