Computing the sides of inscribed polygonsFor regular polygons inscribed in a circle of radius 1, use S ( 6 ) = 1 to conclude that: S ( 12 ) = 2 − 3 S ( 24 ) = 2 − 2 + 3

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Computing the sides of inscribed polygonsFor regular polygons inscribed in a circle of radius 1, use   S  (  6  )  =  1 to conclude that:  S  (  12  )  =      2    −          3        S  (  24  )  =      2    −          2      +              3

mentest91k99 · Accepted Answer

Step 1For a hexagon, each of the six triangles that define the inscribed polygon is an equilateral triangle. Thus, the radius of the circle is 1, and the angle subtended is   π      /    3. To double the number of sides, we half the angle, and use  S  (  12  )  =  2  sin  ⁡            π      12        =  2                    1        −        cos        ⁡                  (          π                      /                    6          )                    2        =      2    −          3      Step 2For 24 sides, half again:  S  (  12  )  =  2  sin  ⁡            π      24        =  2                    1        −        cos        ⁡                  (          π                      /                    12          )                    2        =      2    −          2      +              3

For regular polygons inscribed in a circle of radius 1, use S(6)=1 to conclude that: S(12)=sqrt{2-sqrt{3}}, S(24)=sqrt{2-sqrt{2+sqrt{3}}}.

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