SOLUTION: A circle is circumscribed about a regular hexagon with an apothem of 4.8 centimeters; how do I find the radius of the circle, the length of the side of the hexagon, and the perimet

The apothem divides the bottom side into two equal parts. I will
label each part x. Then the whole bottom side of the hexagon will 
be 2x.  We want to find x so we can find the bottom side 2x.



Draw lines to the vertices



Each of the central angles is 60° because  = 60°.
So all the triangles formed are equilateral, because they are isosceles
because the sides are radii of the same circle, and an isosceles
triangle with one 60° angle is equilateral.  Since all three
sides of an equilateral triangle have the same measure, the bottom 
side is 2x, and so are the radii of the circle, so we mark the radius
at the bottom left 2x.  That means that each side of the
hexagon is the same length as the radius.



 So we only need to look at this one triangle at the bottom.  I'll
erase everything but it.

 

We can use the Pythagorean theorem on that right triangle

   c² = a² + b²
(2x)² = x² + 4.8²
  4x² = x² + 23.04  
  3x² = 23.04
   x² = 7.68
    x = √7.68
    x = 2.771281292 cm

So each side of the hexagon is 2x or 2(2.771281292) = 5.542562584 cm,
and 2x is also the radius of the circle.

Answer: The side and the radius are both 5.542562584 cm.

The perimeter is 6 times the length of side, so the perimeter is
6(5.542562584) = 33.25537551 cm.

Edwin