Surface Area of Cylinder Simplifying Math

hello this is Mr Buffington and we are going to look at the surface area of cylinder what to expect on this we'll talk about the area of circle the area of rectangle and how knowing those two can give us the surface area of cylinder I'd like to introduce you to my round yellow friend this is the circle with circle there are some important parts we have diameter radius the diameter is tce twice as long as the radius it goes across the entire circle through the center radius begins at the center and goes to the outside point on the circle and the area of the circle is the part shaded in in yellow so we're going to use all three parts of those in our lesson today first off let's calculate the area of circle the area of circle equation is this Pi * the radius squared Pi is number that's about 3.1 4 so for our calculations we're going to use the number 3.14 so we will say area is approximately equal to 3.14 * radius squared and to simplify our equation even more we're just going to simplify radius squared to being radius time radius that way we avoid any confusion that might come from using exponents so that's the equation that we're going to use area is approximately equal to 3.1 14 * radius time radius and that's how we got there so let's go ahead and do it find the area when the radius is 9.7 so using our equation all we'll need to do is 3.14 time 9.7 * 9.7 and that should give us 295.27 square cm it's kind of weird number to be working with honestly but that's all right all you're doing is multiplying these numbers you're 3.14 times the radius times the radius so now it's time for us to practice want you to go ahead and solve that one your number's little bit nicer to work with than mine was go ahead and calculate the area when the radius is 7 cm so if you solve that you plug the number seven in where you saw the radius radius time radius and you got 15 53.86 square cm so that's how you find the area of circle that is bit of review from something that we've already done but an important review when we're calculating the area of cylinder our next part is the area of rectangle the area of rectangle is the length times the width so in our example we have rectangle that is three by 15 so we do 3 * 15 and get 45 square cm that's it area of rectangle is very straightforward you take the length of the rectangle times the width that's it your turn go ahead and multiply the length times the width and tell me the area of this rectangle hi welcome back the area would be 7times 3 which is 21 square cm so area of rectangle is very straightforward area of circle is pretty straightforward and what we're going to do now is combine these two to find the surface area of cylinder now I've tried to make an animation that will make this kind of make sense for you so I'm going to cut along the dotted lines and then roll this cylinder out it's kind of funny now I'm going to take the circles from the top and and the circle from the bottom and put them kind of fold them out and then I'm going to get rid of the actual cylinder so notice what you have here is kind of Mickey Mouse on its side sort of thing we have rectangle here and two circles all right rectangle and two circles so if we find the area of the circle and then multiply it times two we'll get both areas of both circles and then you find the area of this rectangle then you can calculate the surface area of cylinder now want to point out that this if we recall if we go back you see that this cylinder all around this circle here that is the length of this rectangle right we rolled it out so it's the same length as the circumference of the end so that's going to come into play here the circumference of the circle is the length of the rectangle the height of the cylinder is the width of the rectangle as soon as we've got that down it then turns into what could be classified as fun or could be classified as just ridiculously complicated here are the four steps that we need to follow to find the surface area of cylinder step one is to find the area of the circles on the top and bottom step two is that we'll find the circumference of the circle that's the distance around the outside of the circle because that will be what we need to find the area of the rectangle then we will add the area of the circles and the area of the rectangle and we will have our final amount so let's do these steps in order first off finding the area of the circle the area of circle equation is pi * radius squar < and we've been using this equation approximately equal to 3.14 * radius * radius the radius of this circle up top is five you can see that up here our radius is five so we'll multiply that out and that gives us the area of the circle on the top which is the same as the area of the circle on the bottom bottom so that step one is done finding the area of the circle step two find the circumference of the circle the circumference is pi * the diameter now this might need little bit of explaining go back to when we thought talked when introduced my friend the circle the beginning radius goes from the center of the circle to the outside the diameter is twice that amount it's the distance completely AC across the circle all right so to find this circumference we multiply pi times our diameter in this case Pi * 10 gives us approximately 31.4 CM that's the circumference we we'll now use the circumference to calculate the area of that rectangle and just like we cut line here and rolled out this cylinder we will get rectangle with the length of the circumference and the width is going to be the height distance here okay remember back to that little silly animation that did okay so when we multiply those we'll get that the area is approximately equal to 3768 square cm I'm always going to be using this approximately equal to sign whenever we move from PI to 3.14 all of our our numbers are going to be approximate all right so our final surface area then step four is that we're going to add the area of the circle 78.5 squ CM we multiply that times two because there's two of them and then we add the surface area of the rectangle 3768 and that should give us an approximate surface area of 5338 square cm that is how you calculate the surface area of cylinder it's got lots of steps but if you follow those steps pretty closely you'll be able to get that surface area so I've got some practice for you one of you solving the surface area of the cylinder go ahead and pause the recording and try this one out and then I'll go through the steps in the order that did them all right step number one area of the circle Pi * the radius time the radius should give you an approximate area of 113.04 square cm that's the area of the circle now I'm going to find the circumference because that will be the length of our rectangle circumference is 3.14 times the diameter remember diameter is different than radius so our diameter is the the full distance across there or twice the radius is going to be so our circumference will end up being 37.68 CM I'm going to use that number when calculating the area of the rectangle that we get by rolling out that cylinder and that will give us 48984 square cm we add that to the two circles have my surface area of the circle multiply that times two because there's two of them and then add that to the 489 and should get my surface area of 75.9 2 square cm that is lot that's ton of stuff you did it though great job you found the area of the circle the area of rectangle and then use those two pieces of information to find the surface area of cylinder hope that lesson was helpful for you have wonderful day