Roots undergo semantic shifts in their meaning. So English "deer" is cognate to German "Tier" despite the fact that Tier means animal in German, but only one specific kind of animal in English. The German meaning is the original one for the root; in English it shifted to the more specific meaning.
In German, "Rad" was originally a word that meant something like "rolling" and it generalized in meaning to encompass what was originally referred to using the cognate of English wheel, which then died out in German. It would have been something like *Wiel (Dutch still retains "wiel").
The evidence for a shared root for wheel goes much much farther back than the split between English and German. The Proto-Indo-Euopean root was something like *kwekwlos, which gives Germanic *hwel (> English wheel), Greek kyklos/cycle, Sanskrit cakra, all according to regular sound changes. So it goes back all the way to the split between these languages.
> Roots undergo semantic shifts in their meaning. So English "deer" is cognate to German "Tier" despite the fact that Tier means animal in German, but only one specific kind of animal in English. The German meaning is the original one for the root; in English it shifted to the more specific meaning.
> In Middle English texts one finds a fish, an ant, or a fox called a der, the Middle English ancestor of our word deer. In its Old English form dēor, the word referred to any animal, including members of the deer family, and continued to do so in Middle English, although it also acquired the specific sense "a deer." By the end of the Middle English period, around 1500, the general sense had all but disappeared.
Wow, I am an absolute layperson, only having listened to the History of English podcast before from where I get some minor context, but I find the whole of linguistics so interesting. Thank you for sharing these details!
In German, "Rad" was originally a word that meant something like "rolling" and it generalized in meaning to encompass what was originally referred to using the cognate of English wheel, which then died out in German. It would have been something like *Wiel (Dutch still retains "wiel").
The evidence for a shared root for wheel goes much much farther back than the split between English and German. The Proto-Indo-Euopean root was something like *kwekwlos, which gives Germanic *hwel (> English wheel), Greek kyklos/cycle, Sanskrit cakra, all according to regular sound changes. So it goes back all the way to the split between these languages.