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

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?

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

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.

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

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

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.

I hope this video has been useful to you.