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It is important to know the rate of intra-molecular contact formation in proteins in order to understand how proteins fold clearly.Here we investigate the rate of intra-molecular contact formation in short two-dimensional compact polymer chains by calculating the probability distribution ρ（r）of end-to-end distance r using the enumeration calculation method and HP model on two-dimensional square lattice.The probability distribution of end-to-end distance p（r）of short two-dimensional compact polymers chains may consist of two parts

i.e.p（r）=p

1

（r）+p

2

（r）

where p

1

（r）and p

2

（r）are different for small r.The rate of contact formation decreases monotonically with the number of bonds N

and the rate approximately conforms to the scaling relation of k（N）∝ N

-α

.Here the value of α increases with the contact radius α and it also depends on the percentage of H（hydrophobic）residues in the sequences of compact chains and the energy parameters of ε

HH

ε

HP

and ε

pp

.Some comparisons of theoretical predictions with experimental results are also made.This investigation may help us to understand the protein folding.