Volume of a Cylinder and Surface Area of a Cylinder

in this video we're going to talk about how to calculate the volume and the surface area of cylinder so let's draw rough sketch of cylinder so this is the radius of cylinder and this is the height of cylinder the volume of cylinder is basically the volume of the base times the height notice that the base of cylinder is circle and represents the area of the base the area of circle is pi so it turns out that the volume of cylinder is just piun time the height it's the area of the base times the height of the cylinder now what about the surface area of cylinder the surface area is going to be the area of the base which is the top Circle and the bottom circle plus the lateral area so it's plus now the area of the base we know the area of circle is piun squ but we have the bottom Circle and the top Circle so it's going to be 2 pi 2 and then the lateral area now if you take let's say piece of paper you can roll it into cylinder let's say this is still the paper but it's about to be rolled into cylinder so the rectangular area of this paper can form the size of cylinder that is the lateral area and we know this is and the length if you turn it into if you roll it up into cylinder that length highlighted in green becomes the circumference of the cylinder so basically it's the perimeter around the circle so therefore it's going to be the circumference 2 pi which represents the length in green times the height so that is the surface area is basically the area of the base that is the area of the circle on top and the bottom plus the lateral area the area that is around the cylinder so those are the two formulas that you need in order to calculate the volume and the surface area of cylinder now let's work on some word problems number one cylinder has radius of 5 cm and height of 10 cm what is the volume of the cylinder so let's draw picture so the radius is 5 cm and it has height of 10 cm so all we need to do is basically plug the info that we have into this formula the volume is pi time the height so it's Pi * the radius squar which is 5^ time the height of 10 5^ is 25 and 25 * 10 is 250 so the volume is Pi so this is the exact answer you can also write the answer this way 250 * pi is 78540 and the units cubic cenm so you can also write the answer like this if you want to number two calculate the surface area of cylinder that has radius of 7 in and height of 12 in so is 7 and is 12 now let's write the formula the surface area is 2i plus 2i * all we need to do is basically plug in everything that we have into that formula so is 7 and is 12 7^ SAR is 49 and 7 * 12 that's 84 2 * 49 is 98 and 2 * 84 is 168 so now what we need to do is add 168 and 98 together 8 + 8 is 16 1 + 6 is 7 7 + 9 is also 16 carry over the one and so this is going to be Pi so that's the surface area if you want an exact answer if you want to round it to the nearest 100th 266 Pi is about this is using the exact answer not 3.14 it's 83566 and the units are inches cubed actually inches squared when you're dealing with volume it's going to be inches Cub centime Cub but when dealing with any type of area or Surface area it's going to be inches squared square feet square yards square centimet things like that so this is the answer you can write it in any of these two ways number three the volume of cylinder is 324 Pi cubic cm if the height of the cylinder is 9 cm what is the surface area of the cylinder so take minute and try this problem it's best to write down the information that you have so we have the volume which is 3 24 pi and we have the height which is 9 cm our goal is to find the surface area of the cylinder now we have the volume equation of the cylinder it's pi 2 time the height and the surface area equation is 2i r^ 2 plus 2i * so we're also missing the radius we need to find before we can find the surface area so let's use this equation to find the value of first the volume is 324 pi and is 9 so the first thing we could do is we can cancel Pi you can divide both sides by pi if you want next let's divide both sides by 9 324 / 9 is 36 so 36 is equal to 2 now all we need to do is take the square root of both sides the square root of 36 is six so now that we have the value of we can now find the surface area so the surface area is going to be 2 pi * 6 + 2 piun * 6 * height of 9 6 SAR is 36 and 6 * 9 is 54 2 * 36 is 72 2 * 54 is 108 72 + 108 is 180 so the answer is 180 Pi now if you want to get the decimal equivalent of that that answer that's going to be 5 65.4 and because we're dealing with area surface area it the units will be square cm number four cylinder has radius of 2.5 ft and height of 14 in what is the volume in cubic in now we need to be careful with the units because right now we have the radius in feet and the height in inches so you don't want to just plug these numbers into the equation because you won't get the right answer the units have to match so let's convert feet into inches because we want the volume in cubic inches 1 foot is equal to 12 in that's our conversion factor so we have 2.5 ft so we're going to multiply by 12 in per foot you want to set it up in in such way that the units feet cancel so 2.5 * 12 is 30 2 * 12 is 24 half * 12 is 6 if you add 24 and 6 that gives you 30 so this is the radius it's 30 in and we already have the height so all we need to do is just use the formula to get the volume so it's Pi * 30 2 * 14 30 * 30 is 900 and 900 time 14 that's 12,600 so it's 12,600 Pi cubic in so that's the volume in this problem for