Cálculo de áreas por el método de Herón

Main ideas / concepts

• Goal: Compute the area of a land plot (a general polygon) using triangulation and Heron’s method (also pronounced “Deron” in the subtitles).

• Heron’s method (core concept):

  • Used to find the area of a triangle when the three side lengths are known.

  • Requires:

    • The triangle’s semi-perimeter: [ s=\frac{a+b+c}{2} ]

    • Then the triangle area: [ A=\sqrt{s(s-a)(s-b)(s-c)} ]

• Why triangulation is used (surveying context):

  • Land plots in surveying are often not triangular or not regular shapes.
  • The polygonal area is approximated by:
    1. Subdividing the polygon into triangles using diagonals.
    2. Applying Heron’s formula to each triangle.
    3. Summing triangle areas to get the total polygon area.
• Precision note:
  • The method is described as not very precise, but it provides a practical approximation of land area.

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Method / step-by-step instructions (detailed)

1. Triangulate the polygon

   • Identify the external vertices of the polygon.
   • Draw diagonals to split the polygon into a set of non-overlapping internal triangles.
   • Keep the triangulation as simple as possible, because:
     • You must apply Heron’s method once per triangle.
   • If the polygon is divided into n triangles, you will calculate n triangle areas.

2. For each triangle (triangle i)

   • 2.1) Determine side lengths
     • Let the triangle’s sides be a, b, c (known from measurements, drawing scale, or trigonometry).
     • Side lengths can be obtained by:
       • Field measurements (directly on the terrain), depending on conditions,
       • Measurements on a scaled drawing, or
       • Trigonometric methods / Law of Cosines, using known angles and sides.

   • 2.2) Compute the semi-perimeter [ s=\frac{a+b+c}{2} ]

   • 2.3) Compute the triangle area using Heron’s formula [ A=\sqrt{s(s-a)(s-b)(s-c)} ]

   • 2.4) Repeat for every triangle in the triangulation.

3. Sum areas to get polygon area [ A_{\text{polygon}} = A_1 + A_2 + \cdots + A_n ]

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Example walkthrough (as presented)

• Given: A topographic polygon (with a “north point” indicated) divided into three triangles: triangle 1, triangle 2, triangle 3.
• Process:
  • Use diagonals to define the three triangles.
  • Determine side lengths for each triangle (a, b, c).
  • Apply Heron’s method separately to each triangle.
• Computed triangle areas (as stated):
  • Triangle 1 area: 2,699.501 square units
  • Triangle 2 area: 6,192.643 square units
  • Triangle 3 area: 5,326.41 square units
• Total polygon area:
  • Sum of the three:
    • 14,952.341 square units (as stated)

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Speakers / sources featured

• No named speakers or external sources are identified in the provided subtitles.
• The content appears to be delivered by an unidentified instructor/narrator (speaking throughout).