What geometric term does the surface of a frozen ice skating rink represent

As, you have used the word surface,

Surface the word is used for two dimensional Object.

Now, Coming to "Surface of a frozen ice skating rink", it is a closed two dimensional curve.

A Closed two dimensional curve Occupies Area.

So, Geometric term which represent surface of a frozen ice skating rink

= Area or Surface Area

Surface = flat area, it doesn't have a height, just an area.