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General Information
Notice that regular polygons have many lines of symmetry. This characteristic is very important in their ability to form
tessellations. As you can see from the figure below, the regular
octagon has eight lines of symmetry: one between each pair of
opposite vertices and one between each pair of opposite sides.
The regular pentagon has five lines of symmetry: one between each
vertex and its opposite side. In general, a regular polygon of
n sides has n axes of symmetry.
All regular polygons with the same number of sides are similar.
In other words, all regular polygons with the same number of sides
can be rotated and enlarged to become congruent to each other.
Let us now consider the interior angles of regular polygons. We know that the three angles in any triangle must sum to 180°. Since regular triangles have angles of equal measure, each angle of a regular triangle must have 60°. (60+60+60=180) What about regular polygons with four sides? The angles of any four-sided polygon (i.e., quadrilateral) must sum to 360°. Since each angle of a regular polygon must have equal measure, each angle of a regular four-sided polygon must have 90°. (90+90+90+90=360) What about regular polygons with five sides? The angles of any five-sided polygon (i.e., pentagon) must sum to 540°. Since each angle of a regular polygon must have equal measure, each angle of a regular five-sided polygon must have 108°. (108+108+108+108+108=540).
In general, what is the common interior angle of an n-sided regular
polygon? It was concluded in the polygons page that the sum of the angles of any n-sided polygon is 180(n-2).
Since all the angles of a n-sided regular polygon must have the
same measure, the common interior angle is 180(n-2)/n.
The measures of the interior angles are extremely important in the exploration of the regular, semiregular, and demiregular tessellations, which are the foundations of tessellations.
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