Hexagon circle radius

A regular hexagon is described and inscribed in a circle. The difference between its areas is 8√3. Find the circle's radius.

Final Answer:

r =  4

Step-by-step explanation:

n=6 Δ=8⋅ 3​=8 3​≐13.8564  Δ=S1​−S2​ r2​=r r = 2r1​⋅ 3​​ r1​ = 3​2⋅ r​   S1​ = 23⋅ 3​⋅ r12​​ S1​ = 23⋅ 3​⋅ 4⋅ r2/3​   S2​ = 23⋅ 3​⋅ r2​  Δ=S1​−S2​ 2⋅  8 ⋅  3​ = 3⋅ 3​⋅ 4⋅ r2/3 − 3⋅ 3​⋅ r2 16 = 4r2 − 3⋅ r2 r2 = 16 r=16​=4   Verifying Solution:  S2​=23⋅ 3​⋅ r2​=23⋅ 3​⋅ 42​=24 3​≐41.5692 r1​=3​2⋅ r​=3​2⋅ 4​≐4.6188 S1​=23⋅ 3​⋅ r12​​=23⋅ 3​⋅ 4.61882​=32 3​≐55.4256  δ=S1​−S2​=55.4256−41.5692=8 3​≐13.8564

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