Quick Calculation for Land Subdivision in GIS Settings

Question: Offset one edge of a quadrilateral (giving a new edge ) to form a new (usually smaller) quadrilateral (parcel) whose area equals a predefined value . Find the offset distance .

Short answer: find the linear equation between offset distance and offset edge , then solve the quadratic equation for the offset :

It is easy to pick the correct root as the solution for .

The linear relationship between and follows from trigonometry:

The quadratic equation of is:

There are other ways to explain .

Let be the origin; , and are vectors, so , , where is the scale ratio.
Then the linear equation relating and can be proved.

From the perspective of similar triangles,

Simplified as:

In a GIS work environment, offsetting a line or reading the length of a line is straightforward and convenient. In order to get the equation of and , a trick can be used here.

Start from the general form:

Let ; then

Offset by a specific value — say 10 — and read from the GIS software; then

The problem is solved without needing angles or the distance , which could be far away.

My practice is to put the logic into an Excel form with three parameters.

Area of the new parcel, in
Length of the original edge, in
Length of the edge after offsetting by , in
Required offset, in B69: =IFERROR((-B67+SQRT(B67B67+2(B68-B67)/10*B66))/((B68-B67)/10),0)