Regular Polygons

 1. General Information 2. Angle Measurements

General Information
Regular polygons are polygons with sides of equal lengths and angles of equal measure:

 Examples of regular polygons

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.

 Eight lines of symmetry of a regular octagon; five lines of symmetry of a regular pentagon

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.

 The pentagon on the left is similar to the one on the right

 The angles of a regular triangle
Angle Measurements
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.

Interior angle measures in regular polygons

 name number of sides sum of interior angles interior angle triangle 3 180 60 square 4 360 90 pentagon 5 540 108 hexagon 6 720 120 heptagon 7 900 128 4/7 octagon 8 1080 135 nonagon 9 1260 140 decagon 10 1440 144 undecagon 11 1800 150 dodecagon 12 2340 156 ... ... ... ... n-gon n 180(n-2) 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.

 Real examples of regular polygons:

 Visit the templates page for templates of regular polygons. You may want to experiment putting regular polygons together to form tessellations before continuing in the site.

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