It's not necessarily the absolute diameter of the projectile, but the amount of kinetic energy behind it, as imparted by the mass and velocity of the projectile (much of the latter given by the size of the propelling charge in the cartridge case) as well as the design of the projectile itself.

The M3 75 mm gun, firing the M61 APC projectile with 2 pounds of powder as propellant, could give the 14.96-pound (6.79 kg) projectile a muzzle velocity of 2,030 feet per second (619 m/s). Using the equation for kinetic energy, KE = 1/2mv^(2), the 75 mm M3 gun could give the projectile 1.30x10^(6) joules of kinetic energy. The M72 plain AP projectile, which largely fell out of use after the North African campaign because of erratic quality and poor performance against German face-hardened armor, has essentially identical ballistic characteristics. The M61 projectile had an energy at the muzzle of 427 foot-tons (the energy needed to raise one long ton, 2,240 pounds, a distance of one foot), while the M72 projectile had a muzzle energy of 398 foot-tons.

The 76 mm M1 gun, firing the M62 APC projectile with 3.75 pounds of powder as propellant, could give the 15.44-pound (7.00 kg) projectile a muzzle velocity of 2,600 feet per second (792 m/s). Using KE = 1/2mv^(2), the M62 APC projectile had 2.20x10^(6) joules of kinetic energy, nearly twice as much as the 75 mm projectile. The M79 plain AP projectile, similar in concept to the M72 projectile, had essentially identical ballistic characteristics. The M62 projectile had a muzzle energy of 724 foot-tons, while the M79 projectile had a muzzle energy of 703 foot-tons. As the 76 mm M1 gun was ballistically matched to the earlier 3-inch M7 gun (used on the M10 tank destroyer), the latter had basically identical characteristics, the only differences being that it fired its projectiles from a different cartridge case and externally retains far more of the appearance of its father, the 3-inch M1918 antiaircraft gun.

This higher kinetic energy gave the 76 mm and 3-inch guns higher penetrating power than the 75 mm gun:

Ammunition|500 m|1,000|1,500|2,000 :--|:--|:--|:--|:--|:-- 75 L31 AP vs RHA|81 mm|66|54|45 75 L31 AP vs FHA|67|52|40|31 75 L31 APCBC vs RHA|72|65|58|52 75 L31 APCBC vs FHA|84|75|67|59

Ammunition|500 m|1,000|1,500|2,000 :--|:--|:--|:--|:--|:-- 75 L40 AP vs RHA|92 mm|76|62|51 75 L40 AP vs FHA|75|58|45|35 75 L40 APCBC vs RHA|81|73|65|59 75 L40 APCBC vs FHA|95|86|79|72

Ammunition|500 m|1,000|1,500|2,000 :--|:--|:--|:--|:--|:-- 76 L52 APCBC vs RHA|116 mm|106|97|89 76 L52 APCBC vs FHA|122|116|110|101 76 L52 HVAP|208|175|147|124

Ammunition|500 m|1,000|1,500|2,000 :--|:--|:--|:--|:--|:-- 3" L50 AP vs RHA|131 mm|107|88|72 3" L50 AP vs FHA|112|92|75|62 3" L50 APCBC vs RHA|115|103|93|84 3" L50 APCBC vs FHA|121|115|107|97

The design of the projectiles themselves can also play a significant part:

Normal uncapped armor-piercing (AP) projectiles perform better against rolled homogeneous armor (RHA) than face-hardened armor (FHA), steel treated to draw the carbon to one side, giving a very hard, thin, outer layer designed to shatter the noses of these projectiles on impact. To combat FHA, as well as armor that was getting progressively thicker and more steeply sloped more effectively than just constantly raising the velocity of normal AP projectiles (making them vulnerable to shattering), armor piercing capped (APC) projectiles were developed. They had a blunt cap of softer metal designed to direct energy away from the nose of the projectile before penetration, hopefully preventing shattering. As the ideal shape of APC projectiles is quite blunt, armor piercing capped ballistic capped (APCBC) projectiles were soon developed, that had an aerodynamic "windshield" on the nose of the projectile.

There is a phenomenon which takes into account the properties of projectiles and contact with armor, known as "shatter gap." If a projectile is too "soft" based upon its Rockwell scale measure (most U.S. and Soviet projectiles fell into this category, but most British and German projectiles didn't), its velocity is increased, and the target armor thickness is held constant, there occurs a point where the projectile nose begins absorbing too much of the impact energy and fails, causing the projectile to break up and potentially refuse to penetrate. For APCBC projectiles, this point occurs when the ratio of projectile penetration over armor resistance (the "true" protective value of the armor) is between 1.05 and 1.25, and projectiles are traveling faster than 2,000 feet per second. For AP shot, the upper and lower limits are a bit higher. Below the range, probability curves apply; above it, all hits will penetrate. This means that projectiles can inexplicably refuse to penetrate at shorter ranges even when tests say they should, but successfully penetrate at longer ranges.

Sources:

Hunnicutt, R. P. Sherman: A History of the American Medium Tank. Novato: Presidio Press, 1978.

Bird, Lorrin R., and Robert D. Livingston. WWII Ballistics: Armor and Gunnery. Albany: Overmatch Press, 2001.

United States. War Department. War Department Technical Manual TM 9-1901 Artillery Ammunition. Washington, D.C.: United States Government Printing Office, 1944.