A regular hexagon & a regular dodecagon are inscribed in the same circle. If the side of the dodecagon is ( 3 - 1 ) and if the side of the hexagon is k 4 , then find the value of k .

We know that sin θ 2 = a 2 r , where a be the side of polygon and θ is an angle taken by one side of inscribed polygon and r be the circum-radius of polygon. So, for dodecagon θ = 360 ° 12 = 30 ° . ⇒ sin 30 ° 2 = 3 - 1 2 r ⇒ r = 3 - 1 2 sin 15 ° = 3 - 1 2 × 3 - 1 2 2 = 2 And for hexagon θ = 360 ° 6 = 60 ° . ⇒ sin 60 ° 2 = k 1 4 2 × 2 ⇒ 1 2 = k 1 4 2 2 ⇒ k 1 4 = 2 ⇒ k = 4

A regular hexagon & a regular dodecagon are inscribed in the same circle. If the side of the dodecagon is ( 3 - 1 ) and if the side of the hexagon is k 4 , then find the value of k .

We know that sin θ 2 = a 2 r , where a be the side of polygon and θ is an angle taken by one side of inscribed polygon and r be the circum-radius of polygon. So, for dodecagon θ = 360 ° 12 = 30 ° . ⇒ sin 30 ° 2 = 3 - 1 2 r ⇒ r = 3 - 1 2 sin 15 ° = 3 - 1 2 × 3 - 1 2 2 = 2 And for hexagon θ = 360 ° 6 = 60 ° . ⇒ sin 60 ° 2 = k 1 4 2 × 2 ⇒ 1 2 = k 1 4 2 2 ⇒ k 1 4 = 2 ⇒ k = 4