What is the distance between the tropics?
Let's start by the quick rule of thumb, I'll follow the way I do it mentally as I think the mnemonics I use could help you too.

First, the tropics are at 23.5° of latitude. And remembering that the original definition of meter is "A ten-millionth of an Earth's quadrant", it means the perimeter of Earth is 40,000 km, that consist on 360° of latitude, then:

$1^\circ = \frac{40,000 \, km}{360^\circ}= 111.11 \, km/^\circ$

Now, the distance between the tropics in a first approximation would be

$\text{Distance} = 2 \times 23.5^\circ \times 111.1 \, km/^\circ = 5221.7 \, km$

Now there are a few sources of error in this calculation. First, the actual latitude of the tropics is 23.43692° (or 23°26′12.9″). Second, the perimeter of Earth is not exactly 40,000 km (it is 40,075.017 km along the equator and 40,007.86 km along a meridian). And third, Earth is not a sphere, therefore not all the degrees of latitude cover the same distance. Taking all those factors into account the actual distance is 5,185.9 km (Calculated using a GIS software).

If you want more accuracy than a few hundred meters, you have to specify the date, because the actual latitude of the tropics oscillates between 22.1 and 24.5 degrees on a 41,000-year cycle. And it is currently changing at a rate of about 0.5 arc seconds of latitude per year, that translates in a displacement of about 14 meters every year.

NOTE: To make the answer more readable to everyone I decided to keep it in metric units (it also makes sense for the mnemonics I use to derive the length of a degree of latitude), so I leave the transformation to imperial units to you.