 Welcome to this module on the total surface area of a cylinder, derivation of formula and application. Mathematics is more interesting when you know why a math factor theorem is true. For most of us, when we look at a formula for the first time it often has very little meaning. All formulas will come to life when we understand how it is derived and how it can be applied to the work we do. In this module you will see the derivation of the total surface area of formula for a cylinder and work through several application problems. Let's start with exploring how the formula is derived. You can think of the total surface area of a cylinder as a lateral surface area, which is the area of the rectangle as you go around the cylinder, plus the area of the circles on the top and bottom. Momentarily ignore the circles on the top and bottom of the cylinder and visualize making a vertical cut down its surface. When this lateral surface is opened up it forms a rectangle. The length of the rectangle is the same as the circumference of the circle. Remember, this formula is pi multiplied by the circles diameter. The width of the rectangle is the height of the original cylinder. To calculate the area of the rectangle we multiply length times width. This means the formula to figure out the lateral surface area or LSA is pi times diameter times height. Now let's return to those circles, the top and the bottom that we ignored earlier. The area of the bases is two times the area of one circle since the cylinder has both a top and a bottom. The formula for calculating this area equals 2 times pi times the diameter squared divided by 4 or simplified to equal pi times the diameter squared divided by 2. Finally, we can calculate the total surface area of the cylinder by adding the lateral surface area from the rectangle to the area of the bases, the top and bottom circle. The formula for the total surface area equals pi times diameter times height plus pi times the diameter squared divided by 2. Let's see this in action. We'll practice these calculations by looking at a variety of objects you'll run into on the job. We'll start with a soup can. How would a manufacturer calculate how much paper is needed to wrap around the can? We'll start with the dimensions. The soup can is 3.625 inches tall and 2.625 inches in diameter. Since the label doesn't cover the top or the bottom we only need to calculate the lateral surface area. Using the lateral surface area formula we multiply pi times the diameter times the height then round our answer to the tenths place. Campbell's uses a label that's approximately 29.9 square inches to cover each soup can. We could also peel the label off and measure by hand the length and width of the resulting rectangle. The length measures 8.5 inches and the width measures 3.625 inches. Plugging those numbers in the area formula for a rectangle area equals length times width we get 30.8125 square inches. Notice that this area is slightly larger than 29.9 square inches. The reason for that is because when the label is put on the soup can there is some overlap. Siloes and cylindrical storage units have been designed to keep bulk products safe. The agriculture, oil and cement businesses are just a few of the many industries that use similar cylindrical storage units. Maintaining their exterior is essential to both improve their aesthetics and prevent deterioration from weathering. You may need to estimate how much it's going to cost to cover the storage cylinder in paint or an industrial coating. When this happens you will always start with figuring out the external surface area. This silo has a diameter of 80 feet and a height of 48 feet. Let's calculate the total surface area that needs covering. For the purpose of this exercise we're assuming no coating needs to be applied to the circular floor but the flat circular exterior top does need a coating. We'll start with the total surface area formula which is the lateral surface area plus the area of the circle on top. We adjust the formula to remove the circular bottom from our calculation. Our starting formula is that the total surface area for this cylinder equals pi times diameter times height plus pi times diameter squared divided by 4. Then we plug in 80 for the diameter and 48 for the height. This gives us 12,063.71579 plus 5,026.548246. When we add those together we figure the total surface area is equal to 17,090.26404 feet squared or approximately 17,090 feet squared. Now it's your turn. Calculate the lateral surface area of a cylindrical tank with an inner radius of 3 feet and a height of 10 feet that needs to be lined with sheet copper. How many square feet of sheet copper are needed? Round the final answer to the nearest tenth of a square foot. You have completed this module on the total surface area of a cylinder derivation of formula and application.