10176

# Reciprocals of Diagonal Lengths in the Regular Polygon of Unit Side

In a regular polygon of unit side, a similar triangle argument shows that the reciprocal of the length of a given diagonal (shown in blue) is the length of the portion of that diagonal to the right of the black vertical diagonal shown. If you number the diagonals in order of increasing size (including the side of the polygon as diagonal one), and if the index of the diagonal is prime to the number of sides of the polygon, the reciprocal length can be found by adding lengths of other diagonals to the original one and subtracting lengths of other diagonals from it. This is illustrated for several -gons with prime. Choose a prime, choose a diagonal, and then move the slider to see which diagonal lengths need to be added and subtracted. The total length of the blue and red diagonals is balanced by the green diagonal, except for the reciprocal portion on the right.

### DETAILS

If and if , then . We choose the smallest positive with this property. If , is the diagonal length. Properties of the sine function yield , , and . Thus the formula represents sums and differences of the diagonal lengths. The proof of this formula depends upon the fact that if , .

### PERMANENT CITATION

 Share: Embed Interactive Demonstration New! Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details » Download Demonstration as CDF » Download Author Code »(preview ») Files require Wolfram CDF Player or Mathematica.

#### Related Topics

 RELATED RESOURCES
 The #1 tool for creating Demonstrations and anything technical. Explore anything with the first computational knowledge engine. The web's most extensive mathematics resource. An app for every course—right in the palm of your hand. Read our views on math,science, and technology. The format that makes Demonstrations (and any information) easy to share and interact with. Programs & resources for educators, schools & students. Join the initiative for modernizing math education. Walk through homework problems one step at a time, with hints to help along the way. Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet. Knowledge-based programming for everyone.