Geometry Part 24 Volume and Surface Area of a Box
hello stranger let's start in this video we are going to talk about how to find the
volume
and surface
area
of boxes also known as rectangular prisms or rectangular parallel pipettes so what is volume
? Volume
is a measure of the capacity of a figure of space so how much um if it's water or air how much of something an object can hold is its capacity when we break down the space that an object contains we break it down into cubes little squares that have the same dimensions on each side, if we look at a cube that is one unit times one unit times one unit, we think of it as a unit to the third power because it has three dimensions, and by the way, you may have ever wondered about this, but the power of three is nicknamed q because of this idea of its relationship to three dimensions, so volume
is generally measured in cubic units. an object that we can also think of as the amount of surface
it would occupy if we were to peel the object off because of the way this object is called a net when we peel the figure off of space and like other area
s surface
area
is measured in square units because we are dividing the area
into squares one by one so to understand how we find the volume
of a box I want you to visualize a box or rectangular prism four units by six units by five units my ultimate goal is to get the volume
of that four-by-six-by-five box, but instead of looking at that first, we're going to look at a layer that's only one unit high or thick and remember to find the volume
we want to figure out how many small cubes will fit in the box. , so we're going to start with how many cubes will fit in a single layer of the box, so I'm going to take these little cubes and line them up, and since the box is six units wide I am going to be able to place six small cubes in the front of that layer and then how many are going to go on the good side we already have one on the side there and since the side is four units long we can put three more for a total of four cubes that they go over the side now i could continue like this but since we have four going down the side and six across it makes a grid so how many little cubes would there be in one layer of this box four times six or 24 of these little cubes each slice of this four by six by five rectangular prism 24 of these little cubic units, but how many slices do we have?![geometry part 24 volume and surface area of a box]()
The height tells us that we have five slices, which means that the total number of little cubes that would fill this box or find the capacity of this rectangular prism will be 5 times the 24 cubic units of each, so our total
volume
is 120 cubic units. Now that number can be found by multiplying the three dimensions, four times six times five, which we can think of as length times width times height, so our volume
of any box is going to be length times width times height, so here's your typical chart, we're going to label the length l the width w and the height h the volume
we just said is length times width times height and by the way another way of thinking about this that will be useful to us in the future is notice the length times the width is actually the area
of the base i'm going to draw on this guy's base it's just like the top area
of this base can is just a rectangle so how do you find the area
?![geometry part 24 volume and surface area of a box]()
You multiply the two dimensions length by width, so in the
volume
formula, the length by width part
is actually what I'm going to label the area
of the base is capital b so you can think of the volume
of a box as if it were the area
of the base times the height or you can just think of it as length times width now let's talk about surface
area
how would we find the surface
area
of this box here's a formula two length width plus two length height plus two height width so let's talk about where each of these terms comes from in the formula twice the length times the width is talking about the bottom and the top of the box the area
of the base it will be the length times the width which is also the area
of the top so if we add those two together we get twice the length times the width.![geometry part 24 volume and surface area of a box]()
Now let's see where the next term comes from. The next term says twice the length times the height. Which sides do you think have an
area
that can be found by multiplying the length? multiplied by the height if you said the front and the back of this box you are correct look here the front of this box has the dimensions length by height but also the back we can't see but back here is a corresponding side that also has length times height area
, so if we're looking for surface
area
, we're going to need twice the length times height to get the wrapping paper the size of the wrapping paper that would go on the front on the back of the box to ge In the area
of the two sides we need to multiply h times w h times w gives us the area
of this side of the box and we can't see it but there is a corresponding side back here that also has the dimensions h times w so that's where that term came from now we have formulas we can use any time we are asked to find the volume
or surface
area
of a box so let's give it a try here we have a rectangular prism aka c aha and we're asked to find volume
in surface
area
so what you want to do is immediately write down your two formulas the volume
of this object and we'll be learning volume
formulas for other objects so it's important to keep those clear what is what the volume
for a box is length times width times height and the surface
area
is twice length times width plus twice length times height plus twice height times width so all we have to do it's plugin it doesn't matter what dimension we assign to which label in the case of a box length I'm going to say it's seven the width is two and the height is three you could have changed the width and length maybe when you did it from either way it will work fine let's find the volume
first so the volume
is length which is seven inches by width which is two inches by height which is three inches now you don't have to write the dimensions but I write it just to remind you that when multip lics inch per inch per inch you get inches cubed and 7 times 2 times 3 is going to be let's see that's 42 cubic inches that's how many little cubes will fit inside or fill that box now let's find the surface
area
to find the surface
area
we're going to use this formula so I need the one with the s for the surface
area
I need to plug in here we're going to have 2 times the length which is 2 times 7 inches times the width 2 inches plus we're going to have 2 times the length which is seven inches times the height which is three inches plus we're going to have twice the right he which is three inches times the width which is two inches now multiplication comes before addition in the order of operations so you are going to multiply everything in each term first 14 times 2 is 28 and note these would be units of square inches inch times inches inch raised to the second power plus 2 times 7 is 14 times 3 will give you 42 square inches plus 2 times 3 it's 6 times 2 it's 12 square inches and then we add them together and we get 82 square inches this is how much wrapping paper of course without any overlap it would take to exactly cover every face of this box.![geometry part 24 volume and surface area of a box]()
I hope this video has been useful to you.