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pyramid and the tetrahedron...

T

tmc

Golden Member
If one of the faces of a regular tetrahedron of side 'a' is glued to one of the triangular faces of an equilateral pyramid of side 'a', then how many faces will the resulting solid have?

edit - ok, side denotes the edge length. i.e., the length of each edge in both the pyramid and tetrahedron is 'a'
 
You mean size 'a'?

Assuming you mean that, and assuming you intend those faces to align, 8.
 
Originally posted by: tmc
added some clarification in the puzzle. try doing it - if you think it is simple.
Click to expand...

Just to make sure I'm on the same page, an equilateral pyramid is the same thing as a tetrahedron? I dunno, it looks like 6 to me. 😵 Maybe I'm missing something obvious.
 
Originally posted by: TuxDave
Originally posted by: tmc
added some clarification in the puzzle. try doing it - if you think it is simple.
Click to expand...

Just to make sure I'm on the same page, an equilateral pyramid is the same thing as a tetrahedron? I dunno, it looks like 6 to me. 😵
Click to expand...

no, they are not the same. it is a pyramid, but the length of each edge is 'a'. it still has 5 sides - 1 square and other 4 equilateral triangles.
 
Pyramid? What style?

A pyramid isn't a shape, it's a structure type.

Is it a 4-face or 5-face pyramid? Counting the bottom as a face, thus... is it a tetrahedron as well, or instead of 3 faces (plus bottom), is it 4 faces (plus bottom)?

If it's essentially two tetrahedrons, then the resulting figure would have 6 faces.
If it's a tetrahedron and a 4-faced pyramid (5 counting bottom, i.e. Egyptian pyramids), then it's 7 faces.

Confused as to how this is a puzzle... unless part of the puzzle is figuring out what type of pyramid it is, considering a pyramid could be built in the shape of a tetrahedron...
 
Originally posted by: tmc
Originally posted by: TuxDave
Originally posted by: tmc
added some clarification in the puzzle. try doing it - if you think it is simple.
Click to expand...

Just to make sure I'm on the same page, an equilateral pyramid is the same thing as a tetrahedron? I dunno, it looks like 6 to me. 😵
Click to expand...

no, they are not the same. it is a pyramid, but the length of each edge is 'a'. it still has 5 sides - 1 square and other 4 equilateral triangles.
Click to expand...

Oh a square pyramid then? Then I guess 5 since it looks like two of the faces line up to form a larger face.

So 4 faces + 5 faces - 2 faces (the two joined together) - 2 faces (the two faces that get joined into a continuous face) = 5 faces?
 
Originally posted by: TuxDave
Originally posted by: tmc
Originally posted by: TuxDave
Originally posted by: tmc
added some clarification in the puzzle. try doing it - if you think it is simple.
Click to expand...

Just to make sure I'm on the same page, an equilateral pyramid is the same thing as a tetrahedron? I dunno, it looks like 6 to me. 😵
Click to expand...

no, they are not the same. it is a pyramid, but the length of each edge is 'a'. it still has 5 sides - 1 square and other 4 equilateral triangles.
Click to expand...

Oh a square pyramid then? Then I guess 5 since it looks like two of the faces line up to form a larger face.

So 4 faces + 5 faces - 2 faces (the two joined together) - 2 faces (the two faces that get joined into a continuous face) = 5 faces?
Click to expand...

bingo!. that is the thing. 5 is the answer. most ppl will get 7. but can you prove it in simple terms why the 2 faces (the two faces that get joined into a continuous face), actually do form a continuous face?
 
Originally posted by: tmc
Originally posted by: TuxDave
Oh a square pyramid then? Then I guess 5 since it looks like two of the faces line up to form a larger face.

So 4 faces + 5 faces - 2 faces (the two joined together) - 2 faces (the two faces that get joined into a continuous face) = 5 faces?
Click to expand...

bingo!. that is the thing. 5 is the answer. most ppl will get 7. but can you prove it in simple terms why the 2 faces (the two faces that get joined into a continuous face), actually do form a continuous face?
Click to expand...

hmmm.. ok I give this puzzle a little more credit now. 🙂 Probably need to spend more time figuring out an elegant proof because all I did was some "mind spatial tricks to convince me they lined up"

Basically I took the square pyramid, added an equalateral triangle to one of the faces so it became like a rhombus (first assume that they did line up, will the resulting shape that you added on be a tetrahedron type of proof), did that with the other side and saw that the vertex joining the two rhombus's at the far end was in the same place and every edge had a length 'a'. But yeah lemme think of a better way to prove it to myself.
 
waitaminute. That can't be true because one vertex of the tetrahedron would be centered over the face of the pyramid that it was glued to. Therefore the sides in question would have to slope at an angle to the sides of the pyramid because the sides of the pyramid only angle towards each other at the top. If you were to extend the sides of the pyramid to infinity they would never meet anywhere but at that one place, but the sides of the tetrahedron meet at a place that's in the plane the pyramid's sides never touch. I could be wrong though, can't really visualize it.
 
Originally posted by: SlitheryDee
waitaminute. That can't be true because one vertex of the tetrahedron would be centered over the face of the pyramid that it was glued to. Therefore the sides in question would have to slope at an angle to the sides of the pyramid because the sides of the pyramid only angle towards each other at the top. If you were to extend the sides of the pyramid to infinity they would never meet anywhere but at that one place, but the sides of the tetrahedron meet at a place that's in the plane the pyramid's sides never touch. I could be wrong though, can't really visualize it.
Click to expand...

A line connecting the top of the pyramid to the top vertex of the tetrahedron is parallel to the ground. So if you extend the sides of the square pyramid, the intersection (at the top) would cross over the top vertex of the tetrahedron....

... if that helps in anyway.
 
It is true.. 5 faces..

And you know that it is true, because it is a fact.

edit: nice puzzle, btw 🙂
 
here is the answer,

imagine two pyramids right next to each other - i.e., the square face of the pyramids is sitting on the ground and the one of the edges of the squares touch each other. a tetrahedron will exactly fit in the gap between the two pyramids.

this proves that the two side faces will actually become one face.

edit- awesome pics YOyoYOhowsDAjello 🙂
 
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