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Volume of rectangles inside rectangles

How many crates can we fit into a boxcar? We'll explain that there's more than one way to solve for the volume in this problem. Created by Sal Khan.

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  • mr pants teal style avatar for user roadrunner⭐
    How do u find the surface area of a triangular prism?
    (28 votes)
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    • female robot grace style avatar for user schroedingercat
      Alot of these answers are giving you the volume, which is how much fits inside a triangular prism, but not the surface area, which is how big the surface is - like if you had to wrap it with wrapping paper, how much paper would you need? I break it down into the same shapes so I know what I've calculated already and don't forget a side. First I take the (Width of the base * Height of the tallest point) for the triangle side, then I +add it to the (Length of the base * Width of the base) for the bottom side, then I +add it to the (Length of the base * Height of the tallest point) for the other side that is not the longest side of triangle side, then I +add it to the (Length of the base * Side that is the hypotenuse (longest side of triangle)). In math, that is SA = wh + Lw + Lh + Ls which will always be the unit of measurement^2, since it is an area and not how long around something is and not a volume.
      (8 votes)
  • aqualine sapling style avatar for user HappySand6
    At Sal wrote 12 after describing the volume of a box instead of 12 to the third.
    (5 votes)
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  • leafers ultimate style avatar for user ChrisCao
    The question is about The Ubas moving from Houston to Egypt, but why do the crates say "china shipping."
    (7 votes)
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  • starky seed style avatar for user 22.jaylon.richburg
    how would you find the area with only the height
    (6 votes)
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  • primosaur tree style avatar for user Connor
    Anybody know the question said something about Houston to Egypt but the picture shows "China Shipping"
    (5 votes)
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    • duskpin ultimate style avatar for user NightmareChild
      Well 1.) China makes crates in bulk, so they could have purchased the crates for cheaper from China Shipping. 2.) The crates would have to travel across the ocean to get from the good old state of Texas to Egypt, so China might have gotten some jumbo-crates to stuff the smaller crates from the boxcar inside. 3.) It was probably just an attempt to make the screen and problem seem more interesting, but Sal overlooked that detail. It might have been too much trouble to find something that specific. *\(^-^)/* idk for sure though...
      (5 votes)
  • orange juice squid orange style avatar for user rick lee
    How do you find the volume of a pyramid inside a rectangle?
    (4 votes)
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  • marcimus pink style avatar for user ❄️Zara ❄️
    how do you find the surface area of a rectangular prism
    (4 votes)
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  • male robot johnny style avatar for user Ember
    I NEED HELP WITH THIS PROBLEM!
    The volume of rectangular prism is same as: The
    Volume of cuboid, Volume of cube, Volume of cone or None of these
    (2 votes)
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    • aqualine sapling style avatar for user siempre la bestia {inactive}
      Hi, Daniel.
      First don't stress out and think it through. The formula to solve for the volume of a rectangular prism is V=lwh. Since a square is just a fancy rectangle you can use the same formula for both. This also goes for volume. A cube is just a fancy rectangular prism. I hope this helps and that should be the answer.
      P.S. I am not a teacher. I am just a friendly, weird teen who LOVES math and also LOVES helping others succeed. :)
      (2 votes)
  • starky seedling style avatar for user CAMERONB
    Can someone explain how to get the sa of a rectangle prism better.
    (2 votes)
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    • mr pink green style avatar for user David Severin
      If you think about a rectangular prism, you have a top and bottom, a left and right side, and a front and back, each of these pairs are congruent to each other. Area of any rectangle is lw. One way would be to do this 2(lw + lh + wh). However, with any prism we have a lateral area around the sides (Ph) and two bases (lw), so the general formula for any prism is SA=Ph + 2B where P is perimeter, h is height, and B is area of base.
      (2 votes)
  • piceratops tree style avatar for user tho
    where did you get the problem? just asking
    (2 votes)
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Video transcript

The Ubas are moving from Houston to Egypt. They pack their belongings in rectangular crates and hire a boxcar to ship the crates across land and sea. The crates are made specifically to fit inside the boxcar with their bases facing down. Each crate has a base 5 meters long by 1.5 meters wide. So let me draw that. So crate is 5 meters long, and 1.5 meters wide, and has a height of 2 meters. So its height might look something like this. So it has a height of 2 meters. So that's each of the crates. And they're designed to fit inside a boxcar. So this is a crate right over here. I'll do my best to draw a crate. And they give us the dimensions of the boxcar. A boxcar is 15 meters long. So let me draw a boxcar here. So it's 15 meters long. Maybe I'll try to make sure I can fit it on the page. So that this whole distance would be 5 meters, and then another 5 meters here, and then another 5 meters here. So that would be 15 meters long. So you could fit three of the crates along an edge like that. And then, they tell us that it is 3 meters wide. So this is 1.5 meters wide. So you could put two of these to get you to 3 meters. Let me draw this so we can see what's going on behind the scenes. So you could go 3 meters wide for a boxcar, and then it is 4 meters high. So each of these are 2 meters high. So you could stack one more. And so you have 2 meters plus 2 meters. This entire distance right over here is going to be 4 meters. And I could draw the rest of the boxcar like this. So there's a couple of ways to think about how many crates you could fit in a boxcar. One way would be just the way that we're doing it right now. We could visualize. How many can you fit in this direction along the length? How many can you fit along the width? And how many can you fit along the height? And essentially, if we multiply those three numbers, we would have counted the number of crates that could fit inside. So you could fit 1, 2, 3 along the length. So that'd be 3. You could fit 2 along the width. 1.5 and 1.5 gets you to 3 meters, so times 2. And then you could fit 2 along the height, so times 2, gets us to 3 times 2 is 6 times 2 is 12. You can fit 12 crates in the boxcar. Now, another way you could have done it is you could say, OK, they're telling us that these are designed to fit. So we really just have to compare the volumes. How many times more is the volume of the box car than the crate? I like doing it this way more, just to make sure that the dimensions actually work out, so that you could actually squeeze these in. Because if the dimensions aren't right, even if the boxcar is 12 times the volume of one of the crates, if the crates don't have the right dimensions, you might not be able to squeeze exactly 12 crates in there. But they're telling us that it is the exact dimensions. So we could figure out the dimensions of the boxcar, then the dimensions of the crate. And then we could figure out how much larger the boxcar is, how many times larger. Let's do the boxcar in this blue color. The boxcar is 15 meters long, 3 meters wide, and 4 meters high. So boxcar volume is equal to 15 in cubic meters. So there's 15 meters times 3 meters times 4 meters. So this is going to be in cubic meters. So this is going to be-- let's see, 15 times 3 is 45. 45 times 4 is 180 cubic meters. That's the boxcar volume. And then what's the volume of the crate? Well, the crate volume-- if we do our math right, it should come out to 1/12 of this, because that's what we just figured out, is 5 times 1.5 times 2. So 5 times 1.5 times 2. Well, 1.5 times 2 is 3, times 5 is 15, so 15 cubic meters. So how many times larger is the boxcar than the crate? Well, what's 180 divided by 15? Well, it's exactly 12. 10 times 15 is 150. And then 2 times 15 is 30. 150 plus 30 is 180. So notice, 180 divided by 15 is 12. So either way, however you think about it-- I find this one to be a little bit easier to kind of just visualize the boxes-- you can fit 12 crates in the boxcar.