Why area of circle and sphere differs?

Answer 1

because a circle is in two dimensions and a sphere is in three

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Take a Basketball ... a sphere ... and cut it in half, right through the center. Then take

one of the halfs, and set the open side down on the table.

The round line where the half-basketball touches the table is a circle, and it has the

same diameter as the original basketball had.

You can see that the circle has less area than the half-basketball has, because the

half basketball covers the whole circle, and still has enough area to curve up away

from the table. There's no way that all of that leather could lie flat on the table

inside of that circle.

If you know how to handle some calculus, you can prove that the half-ball standing

on the table has exactly double the area of the circle where it touches the table.

The circle has less area than the half-sphere that's standing on it, so it definitely

has less area than the whole sphere had before you cut it in half.