Volume of rectangular pyramids using rectangular prisms | Grade 7 (TX TEKS) | Khan Academy

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, 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.

Just 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.

For 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 base, the rectangular base, corresponds to this side, and the height here corresponds to that side. Now for my green one, this base corresponds to this face, while the height corresponds to this height.

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? Well, 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. So here is my pink pyramid, and 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 going to smooth them all out, the lentils fit perfectly. So 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?

All right, let's see. Looks like I've lost a few lentils—that's okay. So now when I smooth them out, it fits, and 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. So the reason why this is important is because we want to go back to our original rectangular prism. Even though the pyramids are different shapes, they all have the same volume since they form a rectangular prism. Altogether, their volumes are each one-third of the total volume of the prism.

To formalize what we just discovered, the volume of a rectangular pyramid, one of these guys, is one-third 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!