05-22-2020, 04:02 PM

Back at school, I'm sure we all learned the following formula for a the area of a triangle:

Area of triangle = ½ x base x height

I think we all made sense of this too: we were shown some kind of geometric explanation (and, if you can't remember, here is a video that should refresh your memory). Then, after learning this, we would have moved on to the volume of a pyramid... which looks very similar, with just one small change:

Volume of pyramid = ⅓ x base x height

Trouble is, I was never shown a simple geometric explanation of this . Sure, there were proofs, but these all depended on calculus or some other kind of limiting process. However, the following video appears to provide an explanation - by dividing a cube into six pyramids, each half the height of the cube:

This is nice: you don't need to understand calculus to make sense of it . In fact, you might even notice that this explanation also works for the "Area of a triangle" formula: instead of dividing a cube into six pyramids, just divide a square into four triangles! However, it still doesn't feel completely satisfactory, for several reasons:

So, does anybody have any ideas for addressing these issues, to get the explanation to work for more different kinds of pyramid? Or do you have any other problems or comments to add?

Area of triangle = ½ x base x height

I think we all made sense of this too: we were shown some kind of geometric explanation (and, if you can't remember, here is a video that should refresh your memory). Then, after learning this, we would have moved on to the volume of a pyramid... which looks very similar, with just one small change:

Volume of pyramid = ⅓ x base x height

Trouble is, I was never shown a simple geometric explanation of this . Sure, there were proofs, but these all depended on calculus or some other kind of limiting process. However, the following video appears to provide an explanation - by dividing a cube into six pyramids, each half the height of the cube:

This is nice: you don't need to understand calculus to make sense of it . In fact, you might even notice that this explanation also works for the "Area of a triangle" formula: instead of dividing a cube into six pyramids, just divide a square into four triangles! However, it still doesn't feel completely satisfactory, for several reasons:

- It only works if the height of the pyramid is half the length (and width) of the base. For example, if the pyramid was 3m tall, and had a 6m x 6m square base, then this explanation would work. However, what if we had a pyramid that was 6m tall, with a 6m x 6m square base? Then, that would break this visual explanation: we couldn't make a cube out of six of those pyramids ! (However, I don't think this is a massive problem: it seems intuitively obvious that doubling the height of a pyramid will also double its volume )

- It only works for pyramids which aren't skewed to one side. See this image: the right pyramid is nice and symmetrical, so this explanation works for those. However, the "oblique pyramid" is skewed off to one side - so, it's useless for making cubes . The formula "⅓ x base x height" does still work for oblique pyramids; however, we can't really explain it in the same way. (Again, this isn't a massive problem: I think most people will be willing to accept that skewing a pyramid like this won't change its volume. After all, if you make a pyramid out of modelling clay, and then you skew it off to one side... then, you still have the same amount of clay . However, it's certainly worth making a note of this)

- It only works for square-based pyramids. Other kinds of pyramids (such as triangular-based pyramids) also obey the "⅓ x base x height" formula, but we can't make a cube out of those . In fact, it's not just pyramids: the "⅓ x base x height" formula also works for cones, but I can't imagine how you would make a cube out of a finite number of cones ! If there's a nice geometric explanation for the volume of a cone, then I expect it'd be very different from this one.

So, does anybody have any ideas for addressing these issues, to get the explanation to work for more different kinds of pyramid? Or do you have any other problems or comments to add?

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Lurker101 Wrote:I wouldn't be surprised if there was a Mega Blok movie planned but the pieces wouldn't fit together.

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