The radius is half the diameter partly as a matter of definition. The radius is the distance from the center to the circle in any direction. The diameter is the distance across the circle from one side to the other through the center. So, as we travel the length of the diameter from the circle to the center (that's one radius length) and from the center back out to the circle on the other side, we have traveled the length of the radius twice. That is to say, the diameter is twice the length of the radius.