The cyCircled AddOn allows to customize the appearance of action buttons of numerous third-party AddOns. In order to enable cyCircled skinning for RingMenu, you need to install a plugin file, which is provided here.
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Copy the RingMenu.lua file provided in this folder (WoW/Interface/AddOns/RingMenu/cyCircledPlugin) into the plugins subfolder of the cyCircled AddOn (e.g., WoW/Interface/AddOns/cyCircled/plugins)
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Open the cyCircled.toc file (located at WoW/Interface/AddOns/cyCircled/cyCircled.toc) with a text editor (e.g. Notepad, TextEdit). Near the bottom, you'll see a list of lines looking like
plugins\XXX.lua
Just below that, add the following new line:
plugins\RingMenu.lua
Save the file and re-start your WoW Client.