How many degrees does it take for a bowling pin to fall down?

Well as it is upright,then laying on the floor..90%.

I think Redrose means what angle of tilt is required before the pin will continue to fall. It the pin was a plain thin cylinder, it would only take a small push to knock it over, However, bowling pins are fat bottomed, and heve a low centre of mass. To get something to fall over, the centre of mass must be moved do it is no longer over the base. According to my calculations, the angle of tilt must be about 15 degrees from the vertical (which is about 75 degrees from the deck). That is using 2 inch for the diameter of the base, and 6 inches for the height above base of the centre of mass. (Standard figures are 2.031", and between 5 60/64" and 5 40/64")
http://www.bowlingball.com/BowlVersity/bowling-pin-specifications

On reflection, I would ignore the result of my calculation until peasy or GM (or any other mathematicians) have tried it out. My scientific calculator is working perfectly except for the display.....

A bowling pin can remain completely perpendicular to the decking when struck with a ball, but when it hits the back -stop even a minuscule degree of deflection can send it on its way to flopping state. Forward motion will force the pin over. If you want the balance angle, this is a completely different calculation.