.. _sphx_glr_api_gallery_geometry_polygon:

Polygon area
----------------------------

Euclid's equal areas, Heron's formula, and Pick's theorem, all checked
against the shoelace formula.


.. raw:: html

  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


.. raw:: html

    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Euclid&#x27;s theory of area (Elements I.35-41) rests on two facts: parallelograms on the same base and between the same parallels are equal in area (I.35), and so are triangles (I.37), each triangle being half of the parallelogram (I.41). Sliding the top side along its parallel line changes every shape but, as the shoelace formula in polygon_area confirms, never the area.">

.. only:: html

  .. image:: /api/gallery/geometry/polygon/images/thumb/sphx_glr_plot_01_euclid_equal_areas_thumb.png
    :alt:

  :doc:`/api/gallery/geometry/polygon/plot_01_euclid_equal_areas`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Euclid's Elements, Book I: equal areas between parallels</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Checks Heron&#x27;s formula against the shoelace formula, then shows why the arrangement matters: for a thin &quot;needle&quot; triangle the textbook form sqrt(s(s-a)(s-b)(s-c)) loses most of its digits, while Kahan&#x27;s rearrangement stays accurate.">

.. only:: html

  .. image:: /api/gallery/geometry/polygon/images/thumb/sphx_glr_plot_02_heron_formula_thumb.png
    :alt:

  :doc:`/api/gallery/geometry/polygon/plot_02_heron_formula`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Heron's formula and needle-like triangles</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Counts the lattice points inside and on the boundary of polygons with integer vertices, and checks Pick&#x27;s formula A = I + B/2 - 1 against the shoelace area.">

.. only:: html

  .. image:: /api/gallery/geometry/polygon/images/thumb/sphx_glr_plot_03_pick_theorem_thumb.png
    :alt:

  :doc:`/api/gallery/geometry/polygon/plot_03_pick_theorem`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Pick's theorem: area from lattice points</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>


.. toctree::
   :hidden:

   /api/gallery/geometry/polygon/plot_01_euclid_equal_areas
   /api/gallery/geometry/polygon/plot_02_heron_formula
   /api/gallery/geometry/polygon/plot_03_pick_theorem