Modeling a dome

I’m trying to model the main shape of a dome like this:

I drew the main dome, and I think the underlying shape is something like object 1, where it is flat on the sides – almost cube’ish.

In SketchUp, I tried doing something like this, but I got a completely rounded dome.

I was going to use the technique described in this Youtube video, but it feels like overkill.

How would you go about modeling this dome?

I used the Solid tools>Intersect in SketchUp Pro with two solid groups:

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I think you’ll find it’s more of a modified Octagon. two long sides out of 8, those have the larger arch.
Here’s a rough version using showing the long and short sides, The octagon is then intersected with a sphere, that you can flatten or stretch.
Oct Dome

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I appreciate the breakdown you gave. Helpful in creating a solution for myself.