A Vietnamese guy at my school said South Asians are a different race? Is this true?

There are no definable races in humans. The USA tries to classify people into races, one of which is "Asian". This is ridiculous. Three fifths of the people of the world are Asian and they live in forty eight countries stretching from Turkey to Timor Leste. Asians have thousands of languages and cultures and come in all shapes and sizes and with every possible skin colour from as pale as any in Europe to as dark as any in Africa. "Asian" is not a race or anything like one.

There are regional differences in appearance between people so Mongolians do not look like Koreans who do not look like southern Chinese who do not look like Vietnamese who do not look like Indonesians who do not look like Burmese who do not look like Nepalese, or Sri Lankans, or Afghans or Georgians or Israelis or Yemenis or Russians etc etc etc.

People of Asia are different nationalities and have different cultures and languages but they are not different races. The regional differences in appearance are superficial and do not constitute races which is why no educated or civilised country has classified its people by "race" for decades.

Martians?

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Yes. I consider them two different "races". Vietnamese people are Southeast Asians. South Asian people are Asian Indians.

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Well personally I don't even think that race is a real thing.

But if you are talking about conventional racial classifications then your Vietnamese friend is right. Vietnamese, Chinese, Japanese, Koreans - what we used to refer to as Orientals until that word became problematic - are "East Asian." South Asian refers to the people of the Indian subcontinent and that region - people from India, Pakistan, Sri Lanka, Bangladesh.