I used SketchUp for parts of a general audience talk last spring. Here’s one where I tried conveying the idea that the 3-sphere can be viewed as the one-point compactification of ordinary 3-space and as the union of two balls.
Of course I first walked the audience through the analogous constructions of lower dimensional spheres, but this one’s more fun. The animated gif above doesn’t capture it that well. Download the model and play with it yourself.