Although your piano's tuning pins may appear square at first glance, chances are they really aren't. Most piano tuning pins are actually tapered slightly toward the tip, with flattened sides. The picture on the right shows a standard tuning pin shape -- the area enclosed by the red line is the flat surface, which appears on all four sides. It tapers to a kind of V shape near the point the string wraps around the pin. Also, the pin widens from the tip to the string hole. The part below the blue line is the lightly threaded shank driven into the wood pinblock (below). A truly square pin, which is very, very rare, would appear with four equal flat sides with no taper. Some 19th century square grand pianos used them, but those pianos are found almost entirely in museums nowadays (square tuning tips are still available for antique instrument restorers).
Most modern pianos made in the past 120 years use a tapered pin and need a Standard No. 2 Star tuning lever tip (shown at left above). This is the default tip on most tuning levers. Odds are, unless it's really rare, this is the tuning tip your piano requires. The star shape has the added advantage of allowing more positions of the tuning lever than a square hole would -- An auto mechanic's wrench will damage the pins-- it isn't tapered to accommodate a piano tuning pin.
Tuning pins also come in several different diameters, but virtually all modern pianos use a Standard No. 2 pin. The number 2 pin is .282 (7.16mm) in diameter at the shank. Some older pianos may have been re-pinned at some point to compensate for a worn pinblock, in which case they may have slightly larger pins such as a No. 3 or No. 4. These pins require a larger tuning tip. Some old British or German made pianos (aka birdcage or overdamper (link leaves site) used a slightly smaller tuning pin, a No. 1 star (7.01mm diameter). These pianos are becoming rarer by the day, but may still be found in antique stores. At the time of this writing, both tuning pins and lever tips are available for this smaller size.
In fact, some really, really rare (and I really, really mean it this time :-) old square grands and other ancient pianos actually used an oblong shaped tuning pin. Though I've never actually seen such a piano in my career, tuning lever tips are still available for them. In fact, a number of speciality tips are available from suppliers on request for some tuning levers, so let me know if you need one.|