red crucian carp has a small head

Difference in body shape: koi has a slightly rounded body, while crucian carp has a slightly flattened body; difference in tail shape:

koi has a short tail

red

crusher carp has a long tail

And there is another notable difference:

koi has a whisker on each side of its mouth, while red

carp doesn't have a whisker.

Carp have no whiskers.