Volume of rectangular pyramids using rectangular prisms

0:01- [Instructor] Now, let's look at a rectangular prism. This is not a cube because we can see that all the sides have different lengths. We have the length and the width and the height, and those are all different. To find the volume of this, I would still multiply the length by the width, by the height, but those would all be different lengths.

0:21Just like we did with our cube, we can also break a rectangular prism into three rectangular pyramids. We know that for a cube, the three pyramids are congruent. But when I take these apart, these definitely don't look congruent to me. If I compare each of them to my rectangular prism, I can see that they are all different.

0:51For example, for my pink pyramid, these bases are the same and the height corresponds to the other length. Now, let's take a look at this yellow one. So now, this, the base, the rectangular base, corresponds to this side, and the height here corresponds to that side.

1:18Now, for my green one, this base corresponds to this face while the height corresponds to this height. And so we can see here that all three of the pyramids do look different, but let's compare their volumes. So how do you think we might figure out if the volumes of these pyramids are the same, or how they might compare with each other?

1:44Well, a simple method is to fill them up with something and actually compare which holds more. So today, we're going to do that and we're going to use lentils. The pink one looks like it might be the smallest to me, so I'm going to start with that one and compare the volumes.

2:02So here is my pink pyramid. I'm going to open up the base and pour in some lentils. Let's see. Let's get it nice and flat. This pyramid now is filled with our lentils. And now, let's pour it into the yellow pyramid. So let's see, when I pour all of the lentils into the yellow pyramid, and I'm gonna smooth them all out, the lentils fit perfectly.

2:50So this tells us that the volume of the pink pyramid and the yellow pyramid are the same. And now, let's check the green pyramid. What do you think? Do you think the volume will be the same? Alright, let's see. Looks like I've lost a few lentils. That's okay. So now, when I smooth them out, it fits.

3:26And so this tells me that the volume of all three pyramids are the same. So we've just seen that the shapes of these pyramids are different. Their bases are different dimensions and their heights are different, but the volumes are the same, and that's really interesting.

3:43So the reason why this is important is because we want to go back to our original rectangular prism. So even though the pyramids are different shapes, they all have the same volume. Since they form a rectangular prism altogether, their volumes are each 1/3 of the total volume of the prism.

4:06To formalize what we just discovered, the volume of a rectangular pyramid, one of these guys, is 1/3 the volume of a rectangular prism with the same base area and height. I hope seeing some visuals makes the formulas make more sense. Thanks for watching, and happy mathing.