This study determines the optimal vascular designs for perfusing engineered tissues. Here, "optimal" describes a geometry that minimizes vascular volume fraction (the fractional volume of a tissue that is occupied by vessels) while maintaining oxygen concentration above a set threshold throughout the tissue. Computational modeling showed that optimal geometries depended on parameters that affected vascular fluid transport and oxygen consumption. Approximate analytical expressions predicted optima that agreed well with the results of modeling. Our results suggest one basis for comparing the effectiveness of designs for microvascular tissue engineering.