Looks like fungus; have you googled lawn fungus pictures?

Circular weak spots are almost always due to a fungus in the soil. At a previous home I had one and did not realize it, because it was at least 30 feet across. You could see it in the Google Earth imagery, but not really on the ground.

I have had very good success eradicating certain fungal issues in St Augustine by using corn meal. Corn meal has the advantage of working even at temperatures higher than 85 degrees. However, what you have may not respond to the corn meal approach. Here is a picture of the disease that cornmeal will work against.

The lesions on the blades of grass will kill the entire blade. If you have Floratam St Augustine in full sun, the grass will die in patches, because Floratam grows so flippin fast. It simply outruns the disease.

You can try cornmeal by applying 2 pounds of any ordinary cornmeal per 100 square feet (that's 10 feet by 10 feet as a quick refresher). Apply to an area larger than the circle to be sure you have it covered. Corn meal will take 3 full weeks to see the improvement, so it is sometimes hard to be patient that long. You can buy cornmeal in 50-pound bags from a feed store near you. Call first to establish availability and price. It should be on the order of $10 per bag or less. Corn meal also makes an average organic fertilizer, so there is going to be benefit to the lawn one way or the other. Again, though, it takes 3 weeks. At the end of 3 weeks you should see improved density, color, and growth where it is applied. As a side benefit, the fungus should/may go away.

By any chance are these spots located in low spots where water would accumulate in a rain? What is your watering schedule? How often and for how long do you water?