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