A square is inscribed in a regular dodecagon--a regular polygon with 12 sides--so that they share 4 vertices, as shown. If the area of the dodecagon is 12, what is the area of the square?
If we inscribed the dodecagon in a circle we can find the radius as
1 = (1/2)R^2 ( sin 360 /12)
1 = (1/2) R^2 ( sin 30°)
1 = (1/4) R^2
4 = R^2
2 = R of circle
Diagonal of square = 4
Side of square = 4/sqrt (2)
Area of square = (4/sqrt (2))^2 = 16/2 = 8
