What distance should I zero my Eotech?

If you zero at 10 yds you will basically be holding under from 11yds out to 45yds. At 45 you'll be just about zeroed again and from there you will need to start holding over. You don't plan on shooting that far, so don't worry about it too much for now. Under 10yds and you will also have to hold over. The shorter the range before your zero, the higher you will hold up to a max of whatever your height over bore is.

These are just basic numbers used off the top of my head as you haven't giving any information to go off of to give set in stone ballistic data. Gun, ammo, velocity, elevation are just a few factors. My recommendation would be to download a ballistic app (Chairgun or Strelok) and plug your data into there to get more accurate data.
 
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Not knowing the airgun or pellet, here are two generic ballistic sheets from Strelok Pro for a slow, medium weight .22 caliber pellet. On the dope chart, looking at the right column, "Elevation, MOA" you will see where the scope zeros. One is with a 10 yard zero and one with a 40 yard zero. Where is says "Elevation, Inches", this is essentially the hold over or under needed at the distance shown.

Because you seem to shoot inside of a small area, it would be best to use a short 10 yard zero, relative to that area. Then all you would have to know is how much, in inches (or portions thereof) that you would have to hold over or under to hit directly where you are aiming.

Strelok Pro is a great phone app for plugging numbers into and learning your dope charts and holdovers.

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