What is meant by average power in AC circuits?
Look at these various scenarios for the power waveform obtained when various phase angles are involved: -



The power waveform resulting from the multiplication of voltage and current sine waveforms is at twice the frequency of either voltage or current and is also sinusoidal. 

Because it's at twice the frequency, if you averaged power over one half of the voltage or current period it still works out to be accurate. However, you should be able to see that if it is averaged over a different period such as a quarter or two-thirds (of voltage/current period) then you can get a significant error.

But, for instance, if you began averaging at the trough of the power waveform and finished at the peak (equivalent to a quarter voltage or current period) then you would get the right answer.