What determines the apparent radius of the rainbow?

It depends on where the sun is. If it is near the horizon (behind you) and in front of you there are water droplets, then you will see a rainbow with a radius (in angular measure) of about 42 degrees, because each water droplet returns a cone of light, whose axis is parallel to the direction to the sun and whose aperture is roughly $2 \cdot 42 = 84$ degrees.

I've never seen better explanations of dozens of phenomena concerning rainbows than in Walter Lewin's lectures.

It depends on the position of the sun in the sky; the higher, the smaller the rainbow radius. Likewise it depends on the temperature of the water droplets and atmosphere; the lower the temperature the smaller the rainbow radius. Also, depends on the impurities in the water droplets; the denser the greater the radius. Also depends on the viewers proximity to the water droplets; the closer, the greater the radius. Also depends on the atmospheric pressure between water droplets (similar to temperature effect). Also depends on the seasonal position of the earth on the ecliptic orbit. Once you have this data you can formulate an approximate radius at a point in time.