Question

Explain why the apothem of a regular polygon must be less than the radius

Answer 1

By definition, the apothem of a polygon goes from the center of this figure to the midpoint of the segment and it forms an angle of 90 degrees with it. 
 The radius of a polygon goes from the center to one vertex. Therefore, they form a right triangle where the hypotenuse is the radius. As you know, the hypotenuse is the longer side of a right triangle, so you can conclude that the apothem of a regular polygon must be less than the radius.