One Japanese cup is exactly 200 millilitres. It is the smallest of the common cups, and it comes with a companion trap: the cup that comes with a rice cooker is not this cup, and the two differ by ten per cent.
About the units
The Japanese cup, 一カップ, was set at 200 ml when Japan standardised recipe measures on the metric system, and it is used throughout modern Japanese cookbooks alongside a tablespoon of 15 ml and a teaspoon of 5 ml — the same values as the European metric measures. The complication is the gō, 合, a traditional unit of about 180.39 ml that was the standard measure for rice and sake. Rice cookers are sold with a plastic cup of one gō, 180 ml, and rice cooker water lines are calibrated to it. Measuring rice with a 200 ml kitchen cup and then filling water to the marked line gives a noticeably wetter result, because the ratio the markings assume no longer holds.
The exact factor
The factor is exactly 200, so a Japanese cup is four fifths of a metric cup and 0.845 of a US cup. The gō relationship is the one worth memorising: 1 gō = 180.39 ml = 0.902 Japanese cups. Sake is also traditionally measured in gō, and a one-shō bottle is ten gō, 1.804 litres — the standard large sake bottle size, which is why it is 1.8 litres rather than a round two.
Where you meet this conversion
Japanese recipes for soups, sauces, dashi and simmered dishes all use the 200 ml cup, and translated Japanese cookbooks frequently convert it silently to "1 cup" in English, which introduces an 18 per cent error against the US cup. Rice preparation is the case where precision genuinely matters, and the reliable method is to use the cooker's own cup and its own water lines together, or to abandon both and weigh the rice while measuring water by volume at a stated ratio.