Surface Area of Rectangular Prisms and Cubes Simplifying Math

hello we're going to look at surface area today of rectangular prisms and cubes so surface area basically is the area on the surface not too helpful as that definition them basically if you wanted to paint solid how much paint would it take to cover it think about the outside and the part that you would need to cover you're not talking about filling that solid up you're talking about just covering the outside surface surface area that's what surface area is thinking of it as painting is kind of good way good way to kind of do that and since picture is worth thousand words here's an example of surface area everything that you see that is green all right not the inside of the box but just the outside of the box now we are going to calculate the surface area of rectangular prisms and cubes and to do that you can do it in two ways one is to draw net now what net is is when you take rectangular prism like this for example and you draw out basically all of the surfaces so if were to take this like if had this on piece of paper or something could fold it on these lines and it would make this rectangular prism all right so that's basically what this is It's net is way to show us all the surface area in flat way so you can draw net and then calculate usually with net you just go ahead and pick one of the large rectangles so for example if were looking at this one would probably take the area of this the one that have there in yellow here I'll make it little thicker find the area of maybe that rectangle there okay and then find the area of this rectangle here and multiply that times two because there's two of them okay so if you're drawing net you can usually find quicker you know quick ways to to find the area or the surface area and pick large sections and kind of solve them all right the other way for rectangular prismas and cubes is to use an equation now the advantage to using net drawing out net is that you can use it for any type of shape and you don't have to memorize any equations the advantage the equation is that it's it's easier to use an equation so here's the equation for finding the surface area of this rectangular prism you can see there it's 2 * the length times the height and is length and height two times the height times the width height width and two times the length times the width basically what you're doing is you're finding the area of each side and then multiplying it times two so you find the area here of the length times the width and you multiply times two because it's in the front and also in the back you find the area of the top and you multiply times two because it's in the top and the bottom you find the area of the side and you multiply it times two because it's on the left and also on the right all right so that's the reasoning behind this two times length times the width two times the height times width two times the length times the width or yeah because you're finding the surface area of each of these rectangles and then you're multiply them time two because there's two sides that will match all right let's go ahead and actually use this equation to find the surface area so here we have rectangular prism we're given the length width height and what we'll do is we're going to substitute those values into this equation now the height is usually pretty obvious because it's how tall the the rectangular prism is as far as length and width some people say the length is the longer some people say the width is the shorter length the shorter measurement other people will say well it's always this will always be the length and this will always be the width it really doesn't matter as long as you're consistent so pick one of these two to be your length and one of them to be your width in this case picked three to be my length all right so three is my length that means everywhere see the length I'm going to plug in the value of three and width would then be two so everywhere plug in or see the I'm going to plug in the value of two it's important to be consistent it doesn't matter if you had switched the length to being two and the width to being three that's fine as long as you're consistent when you put those values in height is usually pretty straightforward all right so plug all those values in 2 * 3 * 5 is 30 2 * 5 * 2 is 20 2 * 3 * 2 is 12 and then add them together 30 + 20 + 12 is 62 now because this is surface area we're measuring in square meters so I'm going to give that units at the end so my final answer is that the surface area of this rectangular prism is 62 square all right again that equation is really helpful and length and width often throw people off so just be consistent pick whichever one you want it really you'll get exactly the same answer all right finding the surface area of cube cube is kind of like rectangular prism only all sides are the same length so you can use the surface area of rectangular prism and we'll see you'll see in just moment that this would lead to lot of repetition but I'm going to do it this way just to to show you that you don't need to memorize new equation all right so my length and my height are both three and three my width and my height three and three my length and my width three and three so basically we're doing doing the same thing three times and 3 * 3 is 9 * 2 is 18 18 18 and add them all together for my final surface area of 54 that should say inches square inches all right 54 square Ines cover it up you can't see it now now there is also this equation that you can use if you want to use the equation for cube it would be six times the side squared so instead of saying 2 * 2 * 2 or 2 + 2 should say plus two which is six we would put that six there instead of saying side times side we'll say side squared so it's exactly the same equation it's little bit more condensed for using cube it's little bit quicker to use this equation but you don't need to memorize second equation you really don't just want to point that out you can use the rectangular prism equation or you can use the Cube equation when it's cube there you have it that's how you find the surface area of both rectangular prisms and cubes