Sounds like it could be ringworm.
Copied and Pasted from a website:
Ringworm is the most recognized fungal disease in goats. It is not a worm, but rather a fungus which usually appears during prolonged periods of very wet weather, often when it is difficult to keep the pens clean and therefore less likely to harbor disease-causing organisms.
Ringworm can be located almost anywhere on the goat's body; its appearance is that of a rounded patch of hair surrounded completely by a hairless ring. Left untreated, it gets bigger and bigger. Ringworm is contagious both to goats and to humans.
Treatment involves donning disposable gloves and thoroughly washing the area with a topical skin disinfectant like Betadine Surgical Scrub. Wipe the cleansed skin surface dry and apply 1% Clotrimazole Cream to the affected area. Repeat this treatment daily for at least two weeks and possibly longer, until the ringworm is gone. While ringworm usually doesn't bother the goat, it can take up to a month to cure.
An even better treatment for many fungal diseases involves the use of 97.8% lime sulphur concentrate diluted and applied in dip form to the goat's body. This product must be obtained from a vet.