A few years ago I posted an article about The Best Way To Brace A Wooden Door Or Gate.  Some folks wanted to know if there was a way to use trigonometry to calculate things like the mitre angle for the brace, so it could be just cut on a mitre saw without a whole bunch of test fitting and shaving.  Well, yes, there is.

If we let:

• bW be the brace Width
• rL be the rail Length
• dBR be the distance Between Rails

then:

• bL ( brace Length ) = sqrt( rL^2 + dBR^2 – bW^2 )
• mA (mitre Angle (in degrees)) = 180/pi * ( pi/2 – arctan( dBR / rL ) – arcsin( bW / sqrt( rL^2 + dBR^2 ) ) )

Note that if you are using a calculator to crunch mA then you need to switch from DEGrees mode to RADians mode.  Arcsin is undefined outside of +/-1 so you can’t use degrees within that function — you need to use radians and then convert the answer back to degrees at the end (which is what the 180/pi multiplier does).  The answer produced by the equation is in degrees.

For those interested in where the above came from, here is what I originally worked out on the back of an envelope, tidied up so that (hopefully) it can be read and understood by others:

Let me know if you spot any mistakes.

Cheerio!