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Find the perimeter and area of the regular polygon, centered at o.

It is a hexagon and the area given is 16 times the square root of 3

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Draw a hexagon centered at origin with a "radius" of 2.  Notice that each point of the hexagon makes an angle pi/3.  Draw a perpendicular bisector of the central angle of one section of the hexagon.  It makes a 30-60-90 right triangle with base of 1.  Each section (there are 12 of them) has an area (1/2)bh =      (1/2)*1*sqrt3 = (sqrt3)/2. 

There are 12 so the area of this hexagon is 6 sqrt3.  The area of the given hexagon is 16 sqrt3.  The ratio of the areas is 8:3, so the ratios of the drawn hexagon must be factored by 8/3.

The base forming 1/12 of the perimeter is 1*(8/3) = 8/3.  (8/3)(12) = 32.  The perimeter is 32 and the area is given as 16 sqrt3.