The "much-loved" female cat was found dead by its owner close to the family home in Carnoustie, Angus, after it had been missing for almost 48 hours.
Staff from animal welfare group the Scottish SPCA examined the previously-healthy seven-year-old pet and discovered a gun pellet lodged in its body.
Police have been carrying out door-to-door inquiries and appealed to anyone with information to contact them.
The cat was found on a grassy embankment on the south side of Barry Buddon Burn, near a footpath that runs by houses on Macdonald Smith Drive.
It was shot some time between 6pm on Thursday and 1pm on Saturday, when the animal was found.