Well, you certainly can do it on GIMP. There's an ellipse tool as well:
![enter image description here](../../I/static/images/438a1221b6e634a0045a116f6f336509b18134dd8186e3f1f5107f086ac79273.png)
It's very similar in selecting to paint.net, but you can also change the area of the selection after selecting it the first time --maybe to make fine adjustments-- by clicking and dragging your mouse over the squares appearing when you hover your mouse over the vertices or rectangles when it's over the edges of the square guide:
![enter image description here](../../I/static/images/74b967c8cb2ded7cb6c7865ed3efb68b6258878c39693395a2d4cbdf26e25449.png)
And zoom in if necessary:
![enter image description here](../../I/static/images/9a88841f7e11d9ce0d4b612b3be1d8fa9d1ebed3b32c4b5c9567bc4cc7391fa5.png)
After that, you can hit Ctrl+I (Invert selection or do "Select" > "Invert Selection" from the menubar to select everything outside the ellipse, then press Delete to remove the unneeded part:
![enter image description here](../../I/static/images/ee2a51f7e4ad162bec396d18c16aa11a2a10bb09a7f0e9df0ea9b8eb8221571c.png)
The checked grey pattern indicates that these parts are transparent. If you don't see this, go to "Layer" > "Transparency" > "Add Alpha Channel" and press delete again without deselecting the current selection.
Now your image is ready to be exported as png :)
Yes, there are lots. You could use the eclipse tool of a paint program to highlight the area you want to keep, inverse and the delete the selection. – Dave – 2013-09-06T10:30:32.717