There are ten airlines offering flights to Taipei. The most direct route isn’t necessarily the cheapest – being flexible is key to finding the lowest airfare rates. For example, try searching by month to see the best time of the week to fly. The cheapest date is currently 19 / Jun / 2019.
You can use the filters on your search to see more flights and find cheaper rates – Taipei has more than one airport. So if you don't need to fly into Taipei Taiwan Taoyuan International (Chiang Kai Shek International) Airport, it's worth checking your other options. The 9599 km flight usually takes about 22 hrs 10 mins.
Flights from Newcastle to are usually most expensive during March. If you're planning on going to Taipei during this time, you could take a look at other airport options. Or you could try to get a last-minute bargain.