Acceleration in circular motion

Hello,
Regarding acceleration in circular motion, my textbook says the total acceleration
of an object traveling in a circular path, can be computed by:

$a = \sqrt{a^2_c + a^2_t}$

and can be proved by pythag. thm.
Can someone help me understand this intuitively?
Thanks.

The component of the acceleration that is perpendicular to the Velocity Vector, $a_c$, is associated with changing the direction of the Velocity Vector (turning the velocity vector without changing its magnitude).
The component of the acceleration that is parallel to the Velocity Vector, $a_t$, is associated with changing the magnitude of the Velocity Vector (speeding up or slowing down, without changing its direction).