Flying from Muscat to Islamabad, Islamabad to Turkey, Turkey to Doha and Doha back to Turkey with Charlie our Shi Tzu, all trips were based upon his weight which was counted on Exces Baggage. The cheapest, around 70 lira, was Dalaman to Doha with Turkish airlines and that was as hand baggage inside the cabin.
Never have wee been charged upon the size of the container.