July 24, 2017

Polygon Angles

Polygon Angles in Geometry Education
Robert F Szalapski, PhD

Every student of Euclidean Geometry learns that the angles of a triangle sum to 180^{0}. Extending these results to other polygons is an excellent way to get students thinking more independently about geometry. This topic is well suited for student discovery. You can read more about it on the Polygon Angles page in the Student/Mathematics/Geometry area. On this page I provide materials that may be used for this lesson. (A colleague and I made use of these files in a course at Nazareth College.)

Students will discover that, for a polygon with N vertices,

\displaystyle\mathrm{sum\; of\;}N\mathrm{\; interior\; angles}=(N-2)\times 180^{0}

Additionally students may explore regular polygons and discover that

\displaystyle\mathrm{one\; angle}=\frac{(N-2)\times 180^{0}}{N}

The files which may be downloaded provide the scaffolding for this lesson of discovery.

The first file is a Geometer’s Sketchpad file with convex polygons including polygons through the hexagon. For each polygon there is a construction plus tables of angle and side measures which automatically update as vertices are dragged about. See screen shots for the quadrilateral and its tables in Figure 1 and Figure 2, respectively.

Figure 1.

A screen shot of a quadrilateral construction from Geometer’s Sketchpad. In the Sketchpad version the vertices may be manipulated changing the angle and length measures.

Figure 2.

A screen shot of a table construction from Geometer’s Sketchpad. In the Sketchpad version the table updates automatically as the vertices of the quadrilateral are manipulated.

Students use the Sketchpad file to fill out a table either on the SmartBoard or in hard-copy form. See Figure 3.

Figure 3.

An interactive table for the SmartBoard for exploring the topic of polygon angles.

The Smart Notebook file in notebook and PDF form are available for download.
The lesson may be restricted to convex polygons and the first formula above, or it may be extended to regular polygons and the second formula above. The final column of the table is used for regular polygons, and the regular polygon constructions and their tables appear only in the second file.

Figure 4.

A screen shot of a square construction from Geometer’s Sketchpad.

Figure 5.

A screen shot of a table construction from Geometer’s Sketchpad. In the Sketchpad version the table updates automatically as the vertices of the square are manipulated.

There are many excellent ways to extend this lesson. Once students have discovered the formulae shown above there is a second table to make predictions. Students may make their own polygon constructions to test their predictions. Other ideas are given in the Notebook file and on the Polygon Angles page in the Student/Mathematics/Geometry area.

 

Download Small Geometer’s Sketchpad File

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Download the Smart Notebook file

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Download the PDF Version

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