"...Taputapuātea marae at Rarotonga, which archaeologists have dated to the 13th century, was built by Tangi'ia who brought the central stone with him from the ancient marae of the same name at Ra'iātea." (Wikipedia, 2022).

"...Tangiia's father was Kau-ngaki, but he was adopted by Pou-vananga-roa-ki-Iva, as was his cousin Tu-tapu—afterwards called Tu-tapu-aru-roa, ..." (Smith, 1899).

"Takitumu [in Rarotonga] was the name of Tangiia's canoe and this name was applied both to the tribe which traces its descent from men who travelled to Rarotonga in that canoe, and to the district which they occupy. Though he was from a chiefly family, Tangiia was not himself a man of high rank." (Crocombe, 1961).

"Takitumu means “the burden is removed.” It is the name that Tangiia, chief of Rarotonga, gave his canoe after his victory over his [half] brother, Tutapu, ..." (Cookislandsvoyaging, 2022).

"...definitive settlement of Rarotonga took place in the thirteenth century with the coming of Karika from Samoa and Tangiia-nui from Tahiti." (Kloosterman, 1976).

Notes
For estimated date formula, "Polynesian scholars who know the race well, is that 25 years is nearer the truth, for the Polynesians married early, and many women come into the genealogies, who as a rule, marry very early. ...1850... ... comparing [Hawaii, Rarotonga, New Zealand] genealogical tables... ...24 generations..." [(1850-1255)=595/25=24] (Smith, 1904). Settlement began around 1230-1280 CE (Bunbury et al., 2022; Wilmshurst et al., 2010). I used: (Katarina Bell's 24th great grandparent's generations+2)x25)=650 years, subtracting -(1854 birth year+25), gave me circa 1229-1255 CE. Tangiia-nui was Whiro te tupua's contemporary [note, Whiro's mother is Tangiia's adopted sister] (Batley, 1973).

Sources

Batley, R. A. L. (1973). Ngati Rangi: Whiro's family at Murimotu, North Island, New Zealand. The Journal of the Polynesian Society, 82(04), 43-354.
https://jps.auckland.ac.nz/document/Volume_82_1973/Volume_82%2C_No....

Bunbury, M.-M.E., Petchey, F., & Bickler, S. H. (2022). A new chronology for the Māori settlement of Aotearoa (NZ) and the potential role of climate change in demographic developments. Proceedings of the National Academy of Sciences of the United States of America., 119(46).
https://doi.org/10.1073/pnas.2207609119

Cookislandsvoyaging. (2022). Takitumu.
https://www.cookislandsvoyaging.org/our-vaka/takitumu/

Crocombe, R. G. (1961, July). Land Tenure in the Cook Islands: Takitumu: the tribe of Tangiia. Victoria University of Welllington.
https://nzetc.victoria.ac.nz/tm/scholarly/tei-CroLan-c1-2.html

Gudgeon, W. E. (2022, July 17). Item 02 - [Cook Islands Maori genealogies, collected by W. E. Gudgeon] [Manuscript]. Pacific Manuscripts Bureau.
https://asiapacific.anu.edu.au/pambu/catalogue/index.php/cook-islan...

Kloosterman, A. M. J. (1976). Discoverers of the Cook Islands and the Names they Gave. Cook Islands Library and Museum.
https://nzetc.victoria.ac.nz/tm/scholarly/tei-KloDisc-t1-body-d13.html

Kokaua, T. (2017). Arera-ukarau-te-ariki-katu-ki-te-rangi-ariki-nui.

Nicholas, H. (1892). Genealogies and historical notes from Rarotonga - Part I. The Journal of the Polynesian Society, 1(1), 20-29.
https://www.jps.auckland.ac.nz/document/Volume_1_1892/Volume_1%2C_N...

Smith, S. P. (1899). Hawaiki: the whence of the Maori, being an introduction to Rarotongan history: Part III. The Journal of the Polynesian Society 8(1), 1-48.
https://www.jps.auckland.ac.nz/document/Volume_8_1899/Volume_8,_No....

Smith, S. P. (1904). Hawaiki: The original home of the maori; With a sketch of Polynesian History. Whitcombe and Tombs Limited.
https://en.wikisource.org/wiki/Hawaiki_The_Original_Home_of_the_Maori

Te Ariki-tara-are, P. (1919). History and traditions of Rarotonga. Part VI. The Journal of the Polynesian Society 28(111), 134-151.
https://www.jps.auckland.ac.nz/document/Volume_28_1919/Volume_28,_N...

Wikipedia. (2022, August 27). Marae.
https://en.wikipedia.org/wiki/Marae

Wilmshurst, J. M., Hunt, T. L., & Anderson, A. J. (2010). High-precision radiocarbon dating shows recent and rapid initial human colonization of East Polynesia. Proceedings of the National Academy of Sciences, 108(5), 1815-1820.
https://doi.org/10.1073/pnas.1015876108