A decent revolver of adequate size (.357 Magnum is on the bottom of that list) is more than capable enough to take deer-sized game efficiently. The problem is if the hunter-shooter has the marksmanship chops to pull it off.

From a braced field position, a revolver shooter getting 6-8 minute of angle groups is doing pretty good. A lousy rifle shooter can get 3-4 MOA.

I once ran a HunterShooter event and had a participant bring a custom Thompson/Center Contender handgun that got honest 1 MOA groups – from the bench. On his first shoot scenario at 27 yards he only hit the vitals on the target twice with five shots, the other shots being two to the paunch and a clean miss. They guy had only shot his handgun off a bench and was useless with it otherwise.

Equipment matters but the hunter-shooter is most important!

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