No, Bras do but jockstraps do not. By "cup" I assume you mean "pouch." All jockstraps are sized by waist alone.
A groin cup is something different than a jockstrap. A groin cup it is a hard protective cup that fits around a guy's sexual organs during sports. A jock strap (athletic supporter) is like tight underwear but is designed as only a waistband with an elastic pouch to hold the testicles close to the body so that they do not jangle or get painfully caught out. As for groin cups, there are only 2 "sizes": one for youth, and one for adults. They do not make adult groin cups in different sizes - the size differences on adult groin cup packages are for waist size only. A cup should completely enclose over the top of the genitals as the cushioned rim rests beside the genitals against the groin (above, under, and in the creases between the legs and scrotum). That way any impact will be transfered from the cup to the pubic bone or perineum, thus protecting the incredibly sensitive "balls" (which are terribly vulnerable, but so important to making sperm and testosterone).
It's not just the groin cup that comes in youth and adult sizes. Jockstraps come in youth sizes as well; the pouch is cut smaller to support boys that are less developed.
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Put it on just like underwear with the pouch covering your penis and testicles. Then slid the cup down into the pouch. DO NOT just put the cup under the pouch since if its not in the pouch the cup can move and your testicles can get under it while you are moving. If you get hit with your testicles outside the cup it will be very very painful. This happened to me in Martial Arts and its a mistake you never make twice!
No. Firstly, jockstraps without cups are only made for men and boys because males have external genitals like testicles that uncomfortably bounce around during sports and require support (cups protect the testicles due to how vulnerable and sensitive they are). Secondly, there are no cups made for female jockstraps either because the products made to protect the vulva and the abdomen of a woman are more like a pad or shield that is formed to cover her between her legs. A "cup" is only needed to fit around something, but females do not have external genitals that hang outside the body the way men's testicles and penises do.
The cup is hard plastic and protects your equipment from a direct hit. Some jockstraps have a pouch to hold the cup in place. Others don't hold the cup, they are just really tight to keep you from bouncing around when you run or workout. A cup is the hard plastic shell you wear over your genitals to protect them from impacts. A jockstrap is a type of underwear for sports. Some jockstraps are made with a pouch to hold the cup in place. Others are just elastic and made support your genitals so they don't bounce, kind of like a sports bra for a guy's parts.