The Hunmin Jeongeum Haerye, the treatise introducing the principles behind the Korean alphabet written by its inventor King Sejong in 1446, explains that this glyph was derived from the "outline of the root of the tongue blocking the throat" (presumably in profile of a person facing left), reflecting the velar nature of the consonant /k/.
Gari Ledyard proposes that Sejong derived ㄱ from the 'Phags-pa letterꡂ(k) by dropping the interior box shape. Ledyard gives evidence that Sejong was inspired by 'Phags-pa for the basic glyph forms, although he changed the shapes of the letters drastically in order to enhance the simplicity and rationality of his script, and the ultimate shape of the letters may indeed have been influenced by that of the speech organs (Ledyard 1997).
In contemporary Korean, when ㄱ (g) is placed on the side, it curves as in the examples of 기 (gi), 길 (gil), 김 (gim), 경찰 (gyeongchal), 건강 (geongang), 건물 (geonmul), 게임 (geim), 개 (gae), 검정색 (geomjeongsaek), 안경 (angyeong), etc. However, when ㄱ (g) is placed on top or bottom, it retains its original look, as in the examples of 고려 (goryeo), 관심 (gwansim), 공부 (gongbu), 구상 (gusang), 권력 (gwollyeok), 금발 (geumbal), 금 (geum), 금융 (geumyung), 박다 (bakda), 먹다 (meokda), 죽다 (jukda), etc. This was done in the modern era to make the character look more balanced.