What is the formula to find the surface of a barrel?

The barrel is probably much like a cylinder. All surface area is the lateral area+the area of the base(s).

In this case:

SA=(d)("pi")(h)+(2)("pi")(r2)

SA is surface area

d is diameter of the circle base

h is the height

r is the radius of the base

What you are doing with this equation is figuring out the circumference x the height, so you get the area of the tall part. Then you take the area of one of the bases and multiply it by 2.