SOLUTION: What is the difference in area between the inscribed circle and the circumscribed circle in a 12-gon of edge length 2cm?

Question 1181238: What is the difference in area between the inscribed
circle and the circumscribed circle in a 12-gon of
edge length 2cm?
Found 3 solutions by MathLover1, Edwin McCravy, ikleyn:
Answer by MathLover1(20850)   (Show Source): You can put this solution on YOUR website!

we need a diagonals across 6 sides (), sides (), sides (), sides() , and sides ()
........given

then radius of circumcircle is:

...........where stands for circumcircle

radius of incircle is:

...........where stands for incircle

the difference in area between the inscribed circle and the circumscribed circle in a 12-gon is

Answer by Edwin McCravy(20062)   (Show Source): You can put this solution on YOUR website!

A regular 12-gon is made of 12 triangles (not drawn to scale), where the angle at the left is 1/12th of 2π or π/6. Draw in the arcs of the two circles and a horizontal line dividing the (isosceles) triangle into two congruent right triangles. The radius of the circumscribed (larger) circle is the hypotenuse h, of the upper right triangle and the radius of the inscribed (smaller) circle is the side adjacent, x, to the π/12 angle. We find the hypotenuse h: We find the adjacent side x: The area of the larger circle is The area of the smaller circle is The difference in the areas is By a well-known identity the expression in parentheses is 1. Therefore the required difference is or . Edwin

Answer by ikleyn(52863)   (Show Source): You can put this solution on YOUR website!
.

    I look into the solution by @MathLover1, and I ask myself :    . . . WHAT it WAS . . . ?

                         . . . No answer . . .

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