A regular hexagon has a radius of 11 inches and a perimeter of 66 inches. Find the apothem.

Here's another way to see this.....if the perimeter = 66, then a side must = 11.

And we have 12 congruent right triangles each having a hypoteneuse of 11 (the radius) and a leg (1/2) the length of a side = 5.5.   And the apothem would form the other leg.

So we have...

apothem^2 = 11^2 - (5.5)^2

apothem^2 = (2^2)(5.5)^2 - (5.5)^2 = (4)(5.5)^2 - (5.5)^2    ....factor out the (5.5)^2......

apothem^2 = (5.5)^2 * (4 -1)

apothem^2 = (5.5)^2 * (3)  ......      take the square root of each side  .....

apothem = 5.5√3  

Just as jboy314 found !!!!

There are a few ways to start thinking about this but here is the one that feels most natural to me.  We note that since a hexagon has 6 sides the sum of its interior angles will be 720 degrees and since there are six of them, each must be 120 degrees.  Now let's draw in the radius and apothem somewhere next to eachother on our figure (you should draw this out so you can see).  When we draw the radius and apothem we have created a right triangle.  The radius bisects a 120 degree angle so the interior angle of the triangle at that vertex is 60 degrees, while the final angle of the triangle is 30.  A 30-60-90 triangle!  Lucky us!  Since the hypotenuse(radius) is 11, the shortest leg of the triangle will be half as much.  5.5 in this case.    in a 30-60-90 triangle it always holds the relationship that the longer leg is  greater than the shorter leg by a factor of sqrt(3).  Therefore your apothem has length 5.5sqrt(3)

Here's a picture of perimeter (side), radius and apothem if it helps you picture it.

Picture borrowed from mathisfun.com: http://www.mathsisfun.com/geometry/regular-polygons.html