Lots of answers here - some correct and some, less so. As some have noted, 1:1 scope means you are in 20 feet of water and let out 20 feet of rode. In other words, the anchor is hanging straight down until it just touches the bottom. Contrary to some answers, triangles do figure into this discussion. It is a simple matter of Pythagorean theorem: a2 + b2 = c2 (a squared + b squared = c squared). In this case, "a" is the depth, and "c" is the rode paid out. If we solve for "b", we know what the maximum swing could be.
Now, in the case of 1:1 scope the math is pretty simple:
a = 1
c = 1
b = ?
... therefore ...
a2 = 1
c2 = 1
... therefore ...
1 + b2 = 1
... therefore ...
b2 = 0 and b 0
Therefore, with a 1:1 scope, there appears to be no swing. This is actually illustrative and not just pointless math. This is because, if it were possible to hang an anchor off of your bow and have it hold while your boat swings in a full circle, we know that the radius of the circle is not zero. It is the length of your boat.
This is important because it is often overlooked when we start looking at more practical scopes such as 5:1 or more. My boat is a 32-footer. So, no matter what I do, I will swing through at least a 64-foot circle.
At a 5:1 scope (pretty much the minimum for most cases) in ... say ... 20 feet of depth, let's repeat the math:
a = 20
c = 100
b = ?
... so ...
a2 = 400
c2 = 10000
... so ...
400 + b2 = 10000
b2 = 9600
b = 98 (rounded)
This makes it seem that I would swing through a circle with a radius of 98 feet (196 ft diameter). But, I actually would swing through 130-ft radius or a circle with a 260-ft diameter. That is a significant difference but one that is sometimes ignored.
Now, for me, the difference is pretty academic because I am not going to drop an anchor within 200-300 feet of anything I don't want my boat to touch. But, if you cut things a bit closer, you should not ignore the length of the boat in making those calculations.
As a tangent, it boggles my mind how close people anchor to each other. Many people place a lot of confidence in the mistaken notion that all boats will swing in harmony and unison. To my mind, no part of one boat's circle should overlap any part of another boat's circle. This is because, around here, currents can be going several knots one direction, with winds blowing in a different direction. And, as we move into shallow waters close to shore, we see currents doing one thing in one place, and something completely different just a few feet away. When I see a power boat drop anchor a few feet from a sailboat, it just about defies comprehension.
I know that many places find this sort of packing "necessary" and it is certainly the norm at some locations. And, in fact, if currents are not a factor and winds are pretty consistent, there isn't much risk to it. But, around here, it is not uncommon to find some boats facing one way and other boats facing a different way (possibly 180 degrees), with varying bearings from their anchors. I truly cannot comprehend why there aren't more issues. Or, maybe there are and I just don't see them because I just don't allow myself to get caught up in that.
But, I swear, if one more powerboater drops an anchor within swinging distance of a mooring buoy, I am going to lose my mind. (Yes, for some reason, it is always a powerboater.)