Question: I have a newbie question about focal length, what it actually means for lenses and how I can translate this into "how close to the target it's going to practically take me" (I don't want to use the term zoom here to avoid confusion, perhaps magnifying power is what describes best what I'm after, basically shots from the distance, not macro).

Here's an example:  

I currently have a Sigma 18-200mm lens (therefore approx. 11x zoom), which I use with Canon EOS 400D and also a compact Sony Cyber-Shot DSC-HX60 with focal length 4.3-129mm (approx. 30x zoom).  

Even though 200mm is obviously more than 129mm the Sony camera takes me closer to the target.  

Looking at zoom factor (30x comparing to 11x) this may seem obvious, but it's less obvious when you compare it to other lenses, e.g. 200-500mm lens, which effectively has "only" 2.5x zoom, but actually takes me closer to the target.

How can I make sense of this when looking at other lenses?  

For example let's consider Tamron 16-300mm, its zoom is obviously approx. 20x, but how can I compare it to Sigma 18-200mm or Cyber-Shot 4.3-129mm? Is it going to take me closer to the target, e.g. magnify objects better then both of them or only better than Sigma, but still not as good as Sony? (ignoring quality of picture of course)

Answer: 
Sony Cyber-Shot DSC-HX60 has a small sensor ([6.16x4.62 mm](http://www.digicamdb.com/specs/sony_cybershot-dsc-hx60/)). When talking about "Sigma 18-200mm lens", you don't mention which camera you use it on, but I guess it's an APS-C sensor (24×16 mm), hence 3.8 times bigger. At constant image size on the sensor, the small sensor will only get a tiny portion of the image, hence the subject will appear 3.8 times bigger on your Sony than with your Sigma lens mounted on an APS-C body for the same focal length.

When comparing focal length with different sensor size, it's better to compare in terms of "35mm equivalent", i.e. multiply by 1.5 for APS-C sensors, and by 5.8 for your Sony camera.
There are several characteristics of an objective which define the sizes of objects:

* the equivalent focal length is the focal length of an objective which yields the same FoV on the 35mm film frame;
* the Field of View (FoV) defines the sizes of object on the recorded image, it is calculated from sensor dimensions and focal length and is defined as the angle between things appearing at opposite corners. The smaller the FoV is the bigger the objects appear. The formulae is[![enter image description here](https://i.stack.imgur.com/2WUTw.png)](https://i.stack.imgur.com/2WUTw.png)where `d` is diagonal and `f` is focal length (in same units, be it meters or millimetres). Alternatively, use `d=43 mm` (the diagonal of the 35mm frame) and the equivalent focal length frequently mentioned somewhere.
* the magnification is commonly used as the biggest achievable magnification at all focal lengths and distances from sensor to the object, it is not directly derived from other characteristics. It may be achieved at both short end and long end of an objective;
* the zoom is the ratio between the smallest FoV and the biggest FoV, the smallest focal length and biggest focal length;
* crop factor is the characteristic of a sensor derived from it's size. It is the ratio between the equivalent focal length and the focal length, the ratio between 35mm film frame diagonal and the diagonal of the sensor.

For example, a 43mm objective used on the 35mm film frame (36x24mm, crop factor 1) will have 60 degree FoV and will have some 1:5 magnification when focused closely (it depends). A camera with 10,75 mm objective and 9x6mm sensor (4x crop) will have the same FoV i.e. the distant objects will occupy the same part of the frame.

You should find the size of the sensor which your camera has to compare the scale of the objects which the combination will yield.