As a horse owner, I have always cut the carrots lengthways - they, and any animal can choke on the rounds. I know the animal sanctuaries check if you are feeding carrots and will not let you feed the ponies and donkeys, nor the goats unless they are cut lengthways.
Funny enough, my goats will eat them chopped lengthways, but offered a full carrot, most will walk away!!
The kids are always fussy about carrots, apples and green feed, and the pigs were when little, until one day, you try again, and they decide they are lovely!!