The surveyor's formula (also called the Shoelace formula or Gauss's area formula) is a handy tool...

  

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The surveyor's formula (also called the Shoelace formula or Gauss's area formula) is a handy tool for computing the area of polygonal regions in the plane. For a triangle, it says the following: Suppose our triangle has vertices (a, b), (c, d), and (e,f) oriented in a counterclockwise manner as (c, d) (e, f) (a, b) Figure 1: A triangle in Figure 1. Then the arca of the triangle is given by (ad + cf + cb) – (bc + de + fa). Prove this. Does this formula work if we orient our vertices in a clockwise manner? If not, what happens? The surveyor's formula (also called the Shoelace formula or Gauss's area formula) is a handy tool for computing the area of polygonal regions in the plane. For a triangle, it says the following: Suppose our triangle has vertices (a, b), (c, d), and (e,f) oriented in a counterclockwise manner as (c, d) (e, f) (a, b) Figure 1: A triangle in Figure 1. Then the arca of the triangle is given by (ad + cf + cb) – (bc + de + fa). Prove this. Does this formula work if we orient our vertices in a clockwise manner? If not, what happens?