Josh has it Height will increase by half the width
(Pedantry alert)
Josh does indeed have it, but it means that height will increase by more than half the width. The inner rim width forms a chord across the circle on which the tyre carcass sits*, and that’s fixed. As you increase the circumference of that circle (ie fit a larger tyre) not only does the circle become wider but its centre also moves away from the chord. So you have the increase in radius (the half-the-width that you mention) plus that additional distance from the centre. The narrower the rim, the more significant that effect (at rim width zero the height increases just the same as the width).
I realise this could really use a diagram 😉
Less top than side clearance suggests relatively narrow internal width rims.
That’s one factor, but the other is simply the shape of the frame. It’s quite common for the positioning of the chainstay bridge to result in less radial than lateral clearance.
Does a tyre’s whole profile get bigger with bigger widths?
Put your mountain bike and your road bike on the ground and look at the bottoms of the wheels. Is your mountain bike rim further from the ground than your road bike’s rim? There’s your answer 😉
* for the purposes of this exercise, assume a tyre is uniformly flexible around its circumference; this isn’t quite true, because the tread is stiffer than the sidewalls, but it’s fine for this discussion and in terms of clearance the tread thickness will counteract the effects on stiffness anyway
