Why is the diameter of a sphere the longest distance on a sphere and not the distance from the center of the sphere to any point on the surface of the sphere?
The diameter of a sphere is indeed the longest distance on a sphere because it is a straight line passing through the center and connecting two points on the surface. To understand why it is the longest distance, we need to consider the geometry of a sphere.
A sphere is a perfectly symmetrical three-dimensional shape in which all points on its surface are equidistant from the center. Any line segment drawn from the center to a point on the surface is referred to as the radius, and it represents the distance from the center of the sphere to that particular point.
For any given point on the surface, it is evident that drawing a straight line from that point through the center, and extending it to the opposite side of the sphere, would connect to another point on the surface. This line segment connecting two points on the surface is a chord, and the longest chord that can be drawn on a sphere is its diameter.
Since the diameter passes through the center of the sphere and connects two points on the surface, it necessarily has to be longer than any other line segment connecting the center to any point on the surface. If any other line segment were longer, it would represent a chord that is longer than the diameter, which contradicts its definition.
Thus, the diameter of a sphere holds the distinction of being the longest distance on the surface of the sphere because it spans from one side of the sphere to the opposite side, passing through its center.