Geometry Part 24 Volume and Surface Area of a Box

Hello, I'm missing, let's start in this video. We're 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

volume

is is a measure of the capacity of a space figure, so how much um, whether it's water or air, how much of something an object can hold is its capacity. When we decompose the space that an object contains, we break it up into small boxes, each having the same dimensions on each side, when we look at a cube, which is one unit by one unit by one unit, we imagine it as a cubed unit because it has three dimensions, and by the way, you may have thought about this before, but the power of three has the nickname q because of this notion of its relation to three dimensions, so it will

Volume

generally measured in cubic units.

We're also going to look at the

surface

area

of an object, which we can also think of as the amount of

surface

area

that would be occupied if we were to pull the object a

part

, because that object is called a mesh when we pull the space figure a

part

, and exactly like other

area

surface

area

is measured in square units. Because we are dividing

area

into squares one at a time to understand how to find the

volume

of a box, I want you to imagine a box, or rectangular prism, that is four by six by five units is large.

My ultimate goal is to get the

volume

of this 4 by 6 by 5 box, but instead of looking at that first, let's look at a layer of it that's only one unit high or thick and remember to find the

volume

, which we want to figure out how many small cubes will fit in the box, so let's start with how many cubes fit in just one layer of the box, so I'm going to take these small cubes and line them up, since the box is six units wide I will be able to fit six small cubes across the front of this one layer and then how many will go down the side?

We already have one there on the side and since the side is four units long we can add three more to the side for a total of four cubes. Now I could go on like this but since we have four going down the side and six going over it making a grid so how many little cubes would there be in a layer of this box four by six or 24 of these little ones Cube each disc of this four by six by five rectangular prism contains 24 of these small cubic units but how many discs do we have well the height tells us we have five discs which means the total number of small cubes that filling this box or finding the capacity of this rectangular prism will be 5 times the 24 cubic units in each, so our total

volume

is 120 cubic units.

This 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 each box is going to be length times width times height, so here is your typical box , we will 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 it which will be helpful for us in the future is to note that the length times the width is actually the

area

of the base that I'm going to draw in the base of this guy.

It's the same as the same

area

as the top This base can is just a rectangle. So how do you find the

area

by multiplying the two dimensions length times width? In the

volume

formula, the length times width

part

is actually what I'm going to call capital b the

area

of the base, so you can think of the

volume

of a box as either base

area

times height, or you can think of it simply as length times width introduce. Now let's talk about the interface. How would we find the

surface

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 in the formula twice the length times the width comes from.

It's about the bottom and the top of the box. The

area

of the base will be length times width, which is also the

area

of the top. So if we add these two together, we get twice the length times the width. Let's see where the next term comes from. The next term says twice length times height which sides do you think have an

area

that can be found by multiplying the length times the height if you said the front and back of this box you are correct. See here, the front of this box, right there, has dimensions of length by height, but also the back, which we can't see, but back here is a corresponding side, which also has an

area

of length by height.

So if we look for the

area

, to get the wrapping paper the size of the wrapping paper that would go on the front of the back of the box, we need twice the length times the height, so to get the

area

of the two sides, we need h Multiplying by w h times w gives us the

area

of this side of the box and um we can't see it, but there's a corresponding side back here that also has dimensions h times w That's where the term comes from. Now we have formulas to use when asked to find the

volume

surface

area

of a box.

So let's try so here we have a rectangular prism also known as a box and we are asked to find the

volume

in the

surface

so immediately write down your two formulas the

volume

for this object and we will

volume

formulas for other objects so it is important that you know exactly which is 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 we have to just plug it in, it doesn't matter which dimension we put on which label on a carton length I will say is seven, width is two and height is three, you might have swapped the width and length if you did it either way , it's gonna be okay.

Let's look for

volume

first, so

volume

is length, which is seven inches times width, which is two inches times height, which is now three inches. I don't have to write the dimensions but I'm just writing it to remind you that if you multiply inches by inches by inches you get inches in cubic inches and 7 times 2 by 3 goes lets see that's 42 cubic inches that is the number of small cubes that would fit in or fill that box. now we want to find the

surface

to find the

surface

we are going to use this formula so i need the one with the s for the

surface

i need to plug in here we are going to have 2 times the length which is 2 times 7 inches times the Width is 2 inches, plus we're going to have 2 times the length, which is 7 inches times the height is three inches, plus we're going to have twice the he ight, which is three inches times the width, which is two inches.

Now multiplication comes before addition in the order of operations so multiply everything in each term first 14 times 2 is 28 and note that these would be units of square inches inches times inches inches to the power of 2 plus 2 times 7 is 14 times 3 equals 42 square inches plus 2 times 3 is 6 times 2 is 12 square inches and then we just add them up and we get 82 square inches So much wrapping paper would of course be needed without overlap to exactly cover all the sides of this box. I hope you found this video helpful, if you did please give it a thumbs up so other students can find the video