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Volume of a Pyramid

April 12, 2013

If you’ve listened to the Wrong, but Useful podcast, you’ll know I’m currently trying to convince myself that the volume of a pyramid is one third of the cuboid that encloses it. A few people have got in contact but here’s where I’m up to with my thoughts so far.

Firstly, I don’t have any diagrams. I can’t find an easy way of drawing them so I’m not going to. See if you can visualise what I’m writing.

Imagine a cube with side length x. Now join the centre of the cube to all 8 vertices creating 6 congruent pyramids consisting of one face of the cube as a base and the centre of the cube as the apex.

The 6 pyramids are identical so the volume of one is (x^3)/6

Pick one of the pyramids (I’m thinking of the bottom one). It is enclosed by a cuboid that is x wide, x deep and x/2 high. The x/2 height is because it’s only going up to the centre: half the height of the cube. The volume of this cuboid is (x^3)/2.

Now, 1/3 of (x^3)/2 is equal to (x^3)/6 so I’m convinced that it’s true in this case.

Now to think about what happens when the apex is not in the centre…

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