Why do disks on planes grow more quickly with radius than disks on spheres? In the book, Mr. Tompkins in Wonderland, there is written something like this:

On a sphere the area within a given radius grows more slowly with the radius than on a plane.

Could you explain this to me? I think that formulas shows something totally different:
The area of a sphere is $4 \pi r^2$, but the area of a disk on the plane is $\pi r^2$.
UPDATE:
Later in this book there is:

If, for example, you are on the north pole, the circle with the radius equal to a half meridian is the equator, and the area included is the northern hemisphere. Increase the radius twice and you will get in all the earth's surface; the area will increase only twice instead of four times if it were on a plane.

Could someone explain me this?

If one interprets the area of a sphere as its surface area, then the formula would be $4\pi r^2$.
The area of a circle is $\pi r^2$.
Therefore the surface area of a sphere grows 4 units when the area of a circle grows 1 unit, which is not slower. I'm sure I'm misinterpreting this, though.
