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If you want to be precise, take the over all measurement from the bottom of your barrel to the top of the objective on the scope. Measure the objective on your scope and your barrel under the objective. Then subtract half of the measurement of your barrel and half of the measurement of your objective from the over all measurement.
But you will find that it doesn't have to be all that accurate. Just holding a ruler beside your rifle will be just fine.
I have done this a few times for ballistic software.
My method is pretty much the same as KOna_stinky's
you need three measurements really
With a micrometer, find out:
1. Diameter of the barrel at the point below the sight.
2. Distance from the top of the barrel to the bottom of the scope.
3. Diameter of the scope
* These three measurements should be at the same point on the rifle.
THEN
1. divide the diameter of the barrel by 2. This gives you the distance from the center of your bore to the top of the barrel (radius);
2. Add #1 to the distance from top of the barrel to the bottom of the scope; then
3. Divide the diameter of the scope by 2. This gives you the distance from the bottom of the scope to the center of the sight (radius).
Add all three together and voila, you have the sight height measurement.
These three measurements with a bit of math will give you the exact distance from bore centre to sight centre. Just make sure they are on the same point on the rifle. I find it easier to do it at the end of the scope bell.
Your eyeball is good enough.
eg from iSnipe using my .223
Sight height at 2.237" at 1000yd=34.43moa up
Sight height at 4.474" at 1000yd=32.51moa up
Notice that's double the sight height.
Sight height at 2.500" at 1000yd=34.21moa up
If you can judge within .250", the difference is .22moa
That said, I know my height is 2.237" because I measured as described in above posts.
I usually remove the bolt and insert cleaning rod, this locates the bore and makes easy to place tape measure in the right spot, through the ejection port. Centreline of scope is just eye-balled, cant be too far off.