Trigonometry help please, plunge depth for a v cutter!

[Jonathan]

V-cutters are generally specified by the included angle, i.e. the angle at the tip. In that case the formula is:

Z=d/(2*tan(a/2))
Where:
Z=depth of cut
d=diameter cut
alpha=tip angle, as above.

So for example, lets say you have this cutter:
4x40°x0, 1mm V-type Solid Carbide Engraving Tool Cutter f. CNC Engraving Machine | eBay

The angle is 40 degrees, so suppose you want to cut 1mm wide:
Z=1/(2*tan(40°/2))=1.37mm

However, there's an error since we've assumed the cutter has a sharp point when in reality it's got a flat, which makes things marginally more interesting, hence why I decided to make this post.

The formula you now need is as follows:
Z=(d-f)/(2*tan(a/2))

Using the same example, the tip flat is 0.1mm so:
Z=(1-0.1)/(2*tan(40/2))=1.24mm

There's also the chance that you're using V-cutters with a radiused tip.
Now the formula you'd need is:
Z=r-(r^2-d^2/4)^0.5, for Z<=2r [Note this is also valid for ballnose cutter]
Z=r+(d-2r)/(2*tan(a/2)), for Z>2r

Where r=tip radius.
e.g. suppose this tool:
3x20°x1mm V-type with radius Engraving Cutter graver HM for CNC engraver machine | eBay

It's 20°, and 1mm tip radius so a=20, r=1. Lets say you want to cut 2.5mm wide:
2.5>2*1, therefore:
Z=1+(2.5-2*1)/(2*tan(20/2))=2.42mm
Suppose you want to cut 1mm wide:
1<2*1, therefore:
Z=1-(1^2-1^2/4)^0.5=0.13mm

Edit: If you don't have a calculator to hand, then using google is a quick way to evaluate it, e.g.
http://lmgtfy.com/?q=1%2B(2.5-2*1)%2...20%2F2+degrees))
You could of course just draw it in a CAD program, but where's the fun in that?