'roller', 'tumbler' has traditionally been a three-dimensional solid

Surface area
Having radius r and altitude (height) h, the surface area of a right circular cylinder, oriented so that its axis is vertical, consists of three parts:
the area of the top base: πr²
the area of the bottom base: πr²
the area of the side: 2πrh
The area of the top and bottom bases is the same, and is called the base area, B. The area of the side is known as the lateral area, L.
An open cylinder does not include either top or bottom elements, and therefore has surface area (lateral area)
L = 2πrh.
The surface area of the solid right circular cylinder is made up the sum of all three components: top, bottom and side. Its surface area is therefore,
A = L + 2B = 2πrh + 2πr² = 2πr(h + r) = πd(r + h),
where d = 2r is the diameter of the circular top or bottom.
For a given volume, the right circular cylinder with the smallest surface area has h = 2r. Equivalently, for a given surface area, the right circular cylinder with the largest volume has h = 2r, that is, the cylinder fits snugly in a cube of side length = altitude ( = diameter of base circle).^[8]
The lateral area, L, of a circular cylinder, which need not be a right cylinder, is more generally given by:
L = e × p,
where e is the length of an element and p is the perimeter of a right section of the cylinder.^[9] This produces the previous formula for lateral area when the cylinder is a right circular cylinder.

Hollow cylinder
Right circular hollow cylinder (cylindrical shell)
A right circular hollow cylinder (or cylindrical shell) is a three-dimensional region bounded by two right circular cylinders having the same axis and two parallel annular bases perpendicular to the cylinders' common axis, as in the diagram.
Let the height be h, internal radius r, and external radius R. The volume is given by
Thus, the volume of a cylindrical shell equals 2π(average radius)(altitude)(thickness).^[10]
The surface area, including the top and bottom, is given by
Cylindrical shells are used in a common integration technique for finding volumes of solids of revolution.^[11]