Cylinders are one of the most commonly used solid shapes in manufacturing and design, especially for liquid containers. Everything from cans and drinking glasses to pipes and barrels are based on cylinders. The volume of a cylinder (the amount of space it occupies or can hold) is a function of the height and diameter of the base. This formula is V = (pi*hd^2)/4, where pi is the constant 3.1415926, h is the height, and d is the diameter (twice the radius).
The surface area of a cylinder is the amoung of two-dimensional space on the outside of the object, and is also a function of height and diameter. There are two ways to calculate the surface area of a cylinder: including both the top and bottom circular regions, or excluding them. The former method is preferred if you need the total area all around the object, while the latter method is better if you just need the area of the region between the top and bottom, such as the label on a can.
Step 1: Find the Area of the Region Between the Top and Bottom: If you were to peel the label off of a cylindrical can, you would notice that it is a rectagle. The height of this rectangle is the height of the cyliner, h. What about the length of this rectangle? Since the label goes all the way around the can, its length is equal to the circumference of the circular bases. This is pi*d. Since the area of the rectangle is length times height, the area of the label is pi*dh. The equivalent formula replacing diamter with radius is 2*pi*hr.
Step 2: Add the Areas of the Bases: If you need the total surface area of the cylinder, you must also add the areas of the top and bottom. Since each circle has a diameter of d, their areas are (pi*d^2)/4. And since their are two of them, the sum is (pi*d^2)/2. If you add this to the expression obtained in Step 1, you get the formula for the entire surface area: pi*dh + (pi*d^2)/2. The equivalent formula replacing diameter with radius is 2*pi*hr + 2*pi*r^2.
If you don’t want to perform the computations by hand, there are several online cylinder calculators that will calculate the volume and surface area when you input the dimensions of the object.