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 20100512 17:06:50 