A tetrahedron and an octahedron

Geometry Level 4

The face of a regular tetrahedron, and the face of a regular octahedron are glued together such that their edges align.

How many faces does the resulting polyhedron have?

8 12 10 9 5 6 11 7

Geoff Pilling by Geoff Pilling , 53, USA

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1 solution

Geoff Pilling
Sep 9, 2019

The octahedron has 8 faces. One is covered and three are extended, (yes the exposed faces of the tetrahedron simply extend three of the octahedron faces to make new diamond shaped faces) leaving us with 7 \boxed{7} faces.

1 Helpful 0 Interesting 1 Brilliant 0 Confused

That's really nice. Is there a way of looking at this that makes it "obvious" that they should fit together? Also, how would you describe the resulting shape? Seems like a sort of ice-cream cone to me...

(By the way, you may want to change the title of the problem - I don't think you meant "octagon"!)

Chris Lewis - 1 year, 9 months ago

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Ooops... Your's right, I meant octahedron... Good catch! Lemme see if I can find a good picture and add it to the solution.

Geoff Pilling - 1 year, 9 months ago

I just use the fact that octahedron's vertices are cube's faces' midpoints and tetrahedron's are cube's alternating vertices, so they must be sloped the same way.

Saya Suka - 1 year, 8 months ago

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