Question: how do airlines measure carry-on luggage?
Answer: "Linear inches" simply means the sum total of your bag's length, width and height. So if your airline allows carry-on bags that measure up to 45 linear inches, and your bag measures 20 inches by 10 inches by 9 inches, its measurement in linear inches is 20 + 10 + 9 = 39 inches, so it would be allowed.