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You can get this done in PowerPoint, then insert the result in Word.
In PowerPoint, convert the SmartArt to shapes using the SmartArt Design>Convert>Convert to Shapes command. Or draw your circles using Insert>Shapes. Overlap 2 circles, then copy and paste those and move them over top of the original shapes. Use Shape Format>Merge Shapes>Intersect. All parts of the circles are deleted except for the area where they overlapped. Color that as you wish to show the overlapping part.
If necessary, align the copies of circles with the merged shape, so it looks like a Venn diagram. Select all the shapes, right-click and choose Save as Picture. EMF format is probably good for your purposes, it's a vector format that can resized without it getting blurry. Then import the graphic into your Word document.
Here is Microsoft's page with more detail about using Merge Shapes.
Hello,
I'm using 365 Word, version 2208. I'm not really sure how to show you what I'm dealing with. I guess I'll screenshot the SmartArt Venn diagram in Word and an image of a Boolean Venn diagram (what I was going for). There's no option in layouts, change colors, or anywhere to only color certain parts.
And it's weird because there's a Boolean Venn diagram on a page ofMS's support site, but we can't replicate that in MS Office 365.
I managed to make a rough Venn in MS Paint and use text boxes to manually input the text, but this ability might be something to consider putting into a future Office update.
Thanks,
K. Walling