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This year marks my tenth year as a teacher of Maths. During that time, I've built up a bank of teaching resources and ideas which I want to share with others. I'll polish them up a bit before uploading them here. Some can be incorporated into lessons and others act as discussion points.

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The majority of posts are focused around Maths, but there are some which look at pedagogy and CPD which can be used in other subjects/settings. Search for lesson resources using the tags at the side. Pedagogy has it's own section.

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Resources can be used in educational settings, including private tuition.

Interior Angles of a Polygon

Triangulation seems to be the go-to approach when deriving the sum of interior angles, which feels intuitive given that we are looking at angles inside the polygon, but there are other ways, some of which might be easier to follow.

How might each of these be used to derive the sum of interior angles of a polygon?


Method 1 - Fan Triangulation

Drawing line segments from one vertex to all other vertices of an \(n\) sided polygon produces \(n-2\) triangles, each of which have angles that add up to 180 degrees. The sum of interior angles is therefore \(180(n-2)\).

Method 2 - Interior Point Triangulation

Drawing line segments from one point inside an \(n\) sided polygon to each of its vertices produces \(n\) triangles each of which have angles that sum to 180 degrees. The sum of the all these angles is \(180n\) but this includes the full turn at the point inside the polygon. Subtract 360 degrees to result in \(180n-360=180(n-2)\)

Method 3 - Summation of Interior and Exterior Angles

Each interior and exterior angle pair add up to 180 degrees. There are \(n\) angles so these add up to \(180n\). The sum of interior angles is this, minus the sum of exterior angles, which is always 360 degrees for a convex polygon. We then get \(180n-360=180(n-2)\) degrees, but without considering any triangulation.

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