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Center-crop an image to a circle or square avatar PNG. Free, runs entirely in your browser — no signup, and nothing you paste in is uploaded.
Centre-cropping to a square is straightforward geometry; the reason to do it before uploading is that different platforms crop differently, and a photo that survives one will lose a chin to another. Cropping yourself means you choose what stays in frame.
Circular avatars have their own trap. The circle is usually applied as a CSS mask over a square image, so the corners are hidden rather than removed — and anywhere the mask is not applied, they reappear. Exporting with real transparency in the corners means the image looks right whether or not the surrounding page cooperates.