Star Polygons

Drawing stars is a fun activity. Once you learn how, you can make some really cool looking stars with any number of points you want.

Also, drawing stars can be related to a number of mathematical concepts, such as:

  • multiples and factoring
  • prime numbers
  • symmetry
  • geometry terms (polygon, vertex, edge)

Vi Hart has a great video about drawing stars. She goes pretty quickly, so I’ve included instructions below.

A star polygon is a figure formed by connecting every qth point out of p regularly spaced points lying on a circle.  The star polygon is denoted by {p,q}. So for example, the 5-pointed star, that we are all familiar with, is a {5,2} star, because it has 5 points (p), and you make it by connecting every 2nd (qth) point.

Web

To create the star polygon:

  1. Start with a circle.
  2. Decide how many points you want your star to have, and place that number of equally spaced dots around that circle.
  3. Choose a starting point.
  4. Draw a line to the point q spaces over.
  5. Repeat until you are back to a starting point.
  6. If you reach a starting point and not all points have been covered, lift the pencil, skip over 1 point, and repeat the process. As shown for the {6,2} star below:

Web

I have a worksheet, with these instructions and pre-printed circles and points, available here.

So, how do these cool stars relate to all those other math concepts?

  • Stars where you can touch all points without lifting your pencil, such as the {5,2} star, are called regular polygon stars. In these cases, p and q are relatively prime, which means they have no common factors. Another example is the {10,3} star. There are no numbers (besides 1) that divide evenly into both 10 and 3.
200px-Star_polygon_10-3

{10,3}

  • In cases that create “asterick” stars (such as {8,4}) or regular polygon stars (the ones where you had to lift your pencil), q is a divisor of p. Or you can say the p is a multiple of q. The {9,3} star consists of 3 regular triangles.
StarPolygon6_3

{8,4}

200px-Star_polygon_9-3

{9,3}

  • The number of lines of symmetry for a star polygon is equal to the number of vertices (points).
76026_5_pt_star_mth

5-pointed star with lines of symmetry

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