Area of a Triangle in Terms of the Taxicab Distance

The Nature of Length, Area, and Volume in Taxicab Geometry

“…In prior research regarding area in taxicab geometry [5,11], the underlying assumption has been that the area of a figure should agree in Euclidean and taxicab geometry, although the formulae to compute the area could be vastly different. Additionally, it appears area has only been investigated in two dimensions.…”

“…A taxicab circle of radius r is composed of four isoceles triangles with area 1 2 r 2 (Proposition 1 of [11]). Therefore, a taxicab cylinder of radius r and height h will have volume…”

“…Papers have appeared that explore length in taxicab geometry in greater detail [9,10], and a few papers have derived area and volume formulae for certain figures and solids using exclusively taxicab measurements [2,5,11]. However, this well-built foundation has left a number of open questions.…”

On the ratio of directed lengths on the plane with generalized absolute value metric and related properties

“…Since the plane with the generalized absolute value metric has distance function different from that in the Euclidean plane, it is interesting to study on the plane with the generalized absolute value metric of topics that include the distance concept in the Euclidean plane. ( [1], [2], [4], [5], [6], [7], [9], [12], [10], [11], [13], [14], [16], [23], [24], [25], [19], [26], [27], [28], [29]) These topics are division point, directed lengths, ratio of directed lengths, Menelaus'es Theorem, Ceva's Theorem and the theorem of directed lines. In this paper, GAM is the abbreviation for the plane geometry with the generalized absolute value metric.…”