For Rabbi Akiva the side of a squared Karpeph must be less that 70 Amot and 2/3. But we know that Sqrt ~ 70.7107. It means that a squared Karpeph with a side of 70.7 Amot is not valid although its area is less than 5000 squared Amot.
Actually, this is the Machloket between Rabbi Akiva and Tana Kama.
The position of Rabbi Akiva is really difficult to understand.
Jonathan, Tal Menache, ISRAEL.
The Ritva explains that the Rabanan had two choices: either to deal with the exact measurement of the Chatzer ha'Mishkan, or to give a more simple measurement which is approximately that of the Chatzer ha'Mishkan (in order that people should not get mixed up regarding exactly how much is the square root of 5000, and giving them a number which is easy to remember). According to Rebbi Akiva, it is preferable to forgo the extra .04 sq. Amos and give a more definitive and easily remembered number such as 70 and 2/3.
Thanks for your answer.
If we look at the Halacha as something that is true and not only a convention, it is hard to understand why R. Akiva will not allow a squared Karpeph with a side of 70.7 amos. Indeed, 70.7*70.7 = 4998.49 which is less than 5000 Amos and this calculation is easy. (Not as the calculation of Sqrt!).
I'm sure Rebbi Akiva looked at the Halachah as something that was true and not only a convention. However, this is not a Gezeiras Hakasuv, which is written in stone, but rather an Asmachta where Chazal saw fit to make the amount of a Karfeph the same amount as the Chatzer ha'Mishkan. Rebbi Akiva could easily say that Chazal saw fit to make the amount approximately like the Chatzer ha'Mishkan, but wanted to make a more easily remembered numer such as 70 and 2/3.
When you say that the calculation of 70.7 is easier or as easy, you are making an assumption which Rebbi Akiva rejected. Logically, one could argue that keeping Shiurim in thirds instead of "tenths" gives one eight less options of error. Additionally, it is possible that people were used to measuring in thirds more than in tenths, which would make Rebbi Akiva correct in his estimation that two-thirds is easier to remember.
All the best,