Pagoda is known as a tiered tower with multiple eaves in Europe but not in Japan. The word pagoda used just for the Burmese style pagoda. Therefore Japanese five-story type pagoda is not recognized by Japanese as pagoda. In Japan, it is called Gojunoto, five-story tower.
These five-storied towers can be seen at many temples throughout Japan. Then, there is a reason to be a five-story tower. It comes from the Five elements story; Earth, Water, Fire, Wind, and Void.
Famous Gojunoto are in Horyuji temple; the world oldest wooden pagoda and in Toji temple.