The area of a regular decagon inscribed in a circle is 10 square units.

What is the area of the regular decagon circumscribed about the circle? How about the area of the regular dodecagon circumscribed about the circle?

Let r be the radius of the circle and
n the number of sides of the regular
polygon.

Area of Inscribed Regular Polygon:

   n*r^2*sin(360/n)/2

Area of Cicumscribed Regular Polygon:

   n*r^2*tan(180/2)

Inscribed Regular Decagon:

   10*r^2*sin(36)/2 = 10

          or

   r^2 = 2/sin(36) ~= 3.4026

Circumscribed Regular Decagon:

   10*[2/sin(36)]*tan(18) ~= 11.0557

Circumscribed Regular Dodecagon:

   12*[2/sin(36)]*tan(15) ~= 10.9407

Posted by Bractals on 2010-05-12 17:06:50