Since this remnant has been expanding for over 300 years at incredibly high speed, by now it is quite large.
Distances to Supernova Remnants are very difficult to determine. The best estimates come from the fast moving knots of material that we can actual SEE moving outward through the sky over a period of years. (Go to Crab Nebula Story where on page 3 you can see a fantastic black and white photo of that object expanding through space over a ten year period). The bright spots in Cas-A are examples of some of these knots. If we know how fast the knots are moving through space, and we know how far they move in angle across the sky, we can compute the distance to them....
(2.1) Suppose a fast moving knot is observed to be moving (again, via the Doppler effect) at 5000 km/sec. , how far does it travel in 10 years?
L= 5000km/sec x 3 x 10**8 seconds = 1.5 x 10**12 km.
(2.2) During that time, suppose the knot is observed to move 3 arc-seconds. What is the distance to object?
L= theta(arc-seconds) x distance to object / 206,265
so distance= 206,265 x L / theta
or d= 206265 x 1.5 x 10**12 km / 3 = about 1 x 10**17 km or 3 kpc.
(2.3) Suppose this knot is now 100 arc-seconds from the center of the remnant. When did the Supernova explode?
If it has been travelling at a constant velocity, and it moves 3 arc-sec in ten years, it moved 100 arc-sec in about t= 100 arc-sec / 3 arc-sec per 10 years = 330 years ago or around 1670 AD.
(or, 3 arc-seconds per 10 years is 30 arc-seconds per 100 years or 100 arc-seconds per 330 years). Drawing a simple picture will help visualize this idea...