Okay, I'm back! Since a single circle was too small, I decided to link four of them together to make a square petal card. First I attempted to do it without a template just by eyeballing the sides. It looked okay cut out, but when I folded it up my angles were a tiny bit off so it wasn't square! So I made a second. This time I cut one side of the circle and ruled up a square to line the other three sides up with it... cut them out, fold them and voila. This one is definitely a usable card, and it's large enough to be viable.
Lastly, I decided to try using an oval tag template to make a smaller flower with more petals. Just cutting the tags alone was a disaster. I didn't even bother to try folding it up because I could tell that it was way too uneven to bother! I might use it for something else one of these days, but as a petal card it's a flop. You really need a defined central shape to line the template up properly. Freehanding just doesn't work!
I was going to give up on the idea but then I remembered I had some stencils... I dug them out and located a small octagon. It was just the right size, so I traced it and away I went. The petals don't quite meet in the center, and it's too small to be a stand-alone card, but I'm really pleased with the effect and I'll use it to embellish another item.