Tennis balls are made of a layer of rubber surrounding a hollow center. Then they put a thin layer of adhesive on the rubber and cover it with a layer of felt (usually yellow). The letters that you may see on the felt are just black ink. The air pressure inside the ball is 12 psi (pounds per square inch) greater than ambient air pressure at sea level (the 'normal' air pressure all around you). The ambient air pressure is 14.7 psi, so all together, the pressure inside the ball is 26.7 psi. I'm not quite sure what you mean by the 'amount of air inside the ball,' but you can figure out how much space is in the ball using the equation for the volume of a sphere (4/3*pi*r^3). A tennis ball that I have here is 2 1/4 inches across, with 1/8 inch thick rubber all around. So the open space inside is 2 inches across (the diameter is 2 inches). The radius (r) is half of that, so r=1 inch. So the volume of the inside of the ball is 4/3*pi (pi=3.14) inches^3 = 4.2 inches^3. (You would say this as 4.2 cubic inches of open space.)
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It really depends on the ball. Two-piece balls have a solid rubber core (heated and pressed into shape) and an exterior covering of thermoplastic (ionomer resin). More exprnsive three-piece balls have either a solid rubber or liquid core with rubber wire wrapped around it to keep it under tension. this keeps the ball hard and stops it from losing its shape. This is given a coating of ionomer, balata rubber or polybutadiene, a special plastic based on petroleum that provides bounce. The latter, however, must be mixed with zinc to strengthen it. Three-piece Golf balls can take 80 manufacturing steps, 32 inspections and 30 days to make one ball, whilst two-piece balls take half the steps and can be made in a day.