Since d and r form two legs of a right triangle, simple trigonometry will tell us that the tangent of the angle a is r / d.
tan(a) = (r/d)
Substituting (r = D/2) and (a = d/2) gives us:
tan(d/2) = [(D/2)/d]
d = 2 tan-1 [(D/2)/d]
Therefore, for the Sun, we can calculate its angular diameter as:
d = 2 tan-1 [(1.392 x 106 km/2)/149,598,261 km] = 0.533° (half a degree)
OK, we know that. But we can use this same procedure to calculate the angular diameter of the Sun from each of the other 7 major planets in our solar system. All we need to do is look up their average distance from the Sun which we can do easily enough.
Mercury d = 2 tan-1 [(1.392 x 106 km/2)/57,910,000 km] = 1.4°
Venus d = 2 tan-1 [(1.392 x 106 km/2)/108,200,000 km] = 0.74°
Mars d = 2 tan-1 [(1.392 x 106 km/2)/227,940,000 km] = 0.35°
Jupiter d = 2 tan-1 [(1.392 x 106 km/2)/778,330,000 km] = 0.10°
Saturn d = 2 tan-1 [(1.392 x 106 km/2)/1,429,400,000 km] = 0.06°
Uranus d = 2 tan-1 [(1.392 x 106 km/2)/2,870,990,000 km] = 0.03°
Neptune d = 2 tan-1 [(1.392 x 106 km/2)/4,504,000,000 km] = 0.02°
How does this look diagrammatically?
The Sun would look a bit smaller in the Martian sky and a bit larger in the Venusian sky. The Sun from Mercury would be frighteningly large (and fry you with temperatures around 800° F). The Sun is much smaller from the outer gas giant planets since they're so much further from the Sun than the inner rocky planets. It would be much brighter than any of the other stars in the sky, but not much larger.
Another interesting tidbit. The Moon has a diameter of 3476 km and is 384,400 km from Earth. Plugging these values into the above equation gives:
Moon d = 2 tan-1 [( 3476 km/2)/384,400 km] = 0.52°
So, even though the Moon is much, much smaller than the Sun, it's also much, much closer and looks about the same size in our sky (half a degree). That's why solar eclipses are possible - the Moon can block our view of the Sun given a favorable orbital geometry.
Trigonometry is cool.