The regular octagon in the ceiling of this cathedral has a radius of 10.5 feet and a perimeter of 64 feet. what is the length of the apothem of the octagon? round your answer to the nearest tenth of a foot.

The apothem of a polygon is a line made from the center to the midpoint of any side of the polygon. The radius, on the other hand, is the line from the center to any corner of the polygon. If you illustrate the problem, you can form a right triangle where the hypotenuse is the radius, and the other two legs are the apothem and half of one side of the polygon. 

Perimeter of octagon = 8*length of each side = 64
length of each side = 8 ft

Therefore,
Apothem = √(10.5^2 - (8/2)^2) = 9.71 ft