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Columbus headed west across the Atlantic expecting to reach India, and initially thought that he had landed there.

His latitude for India was correct, but even if his estimates of distance and the absence of the Americas had also been correct, to reach the real India he would still have had to cross a significant mass of land.

Europeans knew about China and India, but did their knowledge not extend into south-east Asia?

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Europeans knew about China, India, and South-East Asia. In fact, at the time, the Indies referred collectively both to what we would call the Indian subcontinent and South-East Asia.

As can be seen in this reimagining of Toscanelli's 1474 map of the Atlantic Ocean, Columbus thought:

  • Japan (Cipangu) was closer to Europe
  • He could reach the Indies through Japan

These statements were controversial at the time, as the Greeks had shown Earth was much larger. When the inhabitants he met in the Americas did not match prior descriptions of the Japanese, he imagined these inhabitants had originated in the Indies.

Reference: https://en.wikipedia.org/wiki/Christopher_Columbus#/media/File:Atlantic_Ocean,_Toscanelli,_1474.jpg

Atlantic Ocean, Toscanelli, 1474

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    So they thought Japan was a few thousand miles south of its actual location. Interesting.
    – Ray Butterworth
    Commented Oct 17, 2022 at 1:59
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    What's shown is a 19th century reimmagination of the map in question. See posterazzi.com/… for the real deal. It would be nice to be able read the writing in the cartouches
    – kimchi lover
    Commented Oct 17, 2022 at 13:35
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    @RayButterworth - It gets worse. Columbus wanted his idea to work, so he also took the smallest estimate of the size of the earth he could find, meaning he also assumed a much smaller distance between those longitude grid lines than there actually is. He was originally laughed out of the courts of Europe for a damn good reason.
    – T.E.D. ♦
    Commented Oct 17, 2022 at 16:02
  • @T.E.D. Columbus problem was that he used the Italian/Roman Mile (1,479 meters) instead of the Arabic mile (1,973 meters) for the 56⅔ miles per degree calculation. See: What did Native Americans know, or speculate, about the Old world? - History Stack Exchange
    – Mark Johnson
    Commented Oct 18, 2022 at 9:10
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    @MarkJohnson - I disagree on a technicality. Columbus' problem was that he was using the "unscientific method": deciding what you want to be true ahead of time, and then go out looking for evidence that might back that up. The mile thing was merely the rotten fruit of his method.
    – T.E.D. ♦
    Commented Oct 18, 2022 at 13:41
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The third book of Marco Polo's travels deals with South East Asia in chapters 5 to 11. Chapter 5 deals specifically with the South East Asian mainland. So Europeans actually had first-hand accounts of some of those places, plus a lot of hearsay.

Re. "India": It is not really clear to me whether Marco Polo considers South East Asia part of India or not. With Yule's translation, one could clearly come away with the impression that for Polo, even Cipangu/Japan is part of India:

Now that I have told you about the ships which sail upon the Ocean Sea and among the Isles of India, let us proceed to speak of the various wonders of India; but first and foremost I must tell you about a number of Islands that there are in that part of the Ocean Sea where we now are, I mean the Islands lying to the eastward. So let us begin with an Island which is called Chipangu.

(Book three, Chapter 1)

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