contains two elements, it has a rectangle as a cylindric section, otherwise the sides of the cylindric section are portions of an ellipse. Finally, if a plane contains more than two points of a base, it contains the entire base and the cylindric section is a circle. In the case of a right circular cylinder with a cylindric section that is an ellipse, the eccentricity of the cylindric section and semi-major axis of the cylindric section depend on the radius of the cylinder and the angle between the secant plane and cylinder axis, in the following way: Volume If the base of a circular cylinder has a radius and the cylinder has height , then its volume is given by . This formula holds whether or not the cylinder is a right cylinder. This formula may be established by using Cavalieri's principle. In more generality, by the same principle, the volume of any cylinder is the product of the area of a base and the height. For example, an elliptic cylinder with a base having semi-major axis , semi-minor axis and height has a volume , where is the area of the base ellipse (= ). This result for right elliptic cylinders can also be obtained by integration, where the axis of the cylinder is taken as the positive -axis and the area of each elliptic cross-section, thus: Using cylindrical coordinates, the volume of a right circular cylinder can be calculated by integration over Surface area Having radius and altitude (height) , 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: the area of the bottom base: the area of the side: The area of the top and bottom bases is the same, and is called the base area, . The area of the side is known as the , . An open cylinder does not include either top or bottom elements, and therefore has surface area (lateral area) . 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, , where is the diameter of the circular top or bottom. For a given volume, the right circular cylinder with the smallest surface area has . Equivalently, for a given surface area, the right circular cylinder with