How does a cell sense its size?
This is a question that has been the focus of study for the last century (e.g., Amodel of cell size regulation - Ycas et al, J. Theoret. Biol. (1965) 9, 444-470). Cell size regulation may be in part determined by ribosomal activity (through mTor regulation) and is a critical checkpoint in cell division.

How the cell senses its size, however, is not understood. In 2009, two reports suggested that protein gradients could be responsible for the sensing of cell size. You can read a commentary about those articles in Cell size control: governed by a spatial gradient. - Almeyda and Tyers, Dev. Cell. (2009) 17(1), 3-4:


  The phenomenon of cell size homeostasis, whereby cells coordinate
  growth and division to maintain a uniform cell size, has been an
  outstanding issue in cell biology for many decades. Two recent studies
  in Nature in fission yeast demonstrate that a gradient of the polarity
  factor Pom1 is a sensor of cell length that determines the onset of
  Cdc2 activation and mitosis.


These articles demonstrate one way in which cells may sense their size, but most probably several other mechanisms are also in place.