Does the Euclid extension require SketchUp pro (paid version)?

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For SketchUp 2018, yes, Euclid requires the Pro version. This is because the Pro version is still a desktop application, which SketchUp has been for years, while the 2018 SketchUp Free version is a web application. As a result, the 2018 Free version doesn't work with Euclid.

For SketchUp 2017, as well as most SketchUp version before that, both the Free and Pro versions will work with Euclid. You can read more about SketchUp and EnergyPlus compatibility on the Release Notes section of the Euclid project page.