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A New Solid (Posted on 2004-10-05) Difficulty: 3 of 5
A regular tetrahedron has four equilateral triangles as faces. A regular square pyramid has four equilateral triangles and a square as faces. The faces of the tetrahedron are congruent to the triangular faces of the square pyramid.

A new polyhedron is created by gluing the tetrahedron and the square pyramid together at a triangular face so that the vertices of the triangles coincide. How many faces does this polyhedron have?

  Submitted by Brian Smith    
Rating: 4.1250 (8 votes)
Solution: (Hide)
At first one might guess 7 faces as 2 faces vanish.
But the polyhedron has only 5 faces as two rhombics are created.
To see this place two pyramids as shown in the figure. (ABCDX and CDEFY)
A   D   E
  X   Y
B   C   F
We know CX=DX=CD=CY=DY. Since CDEFY is a translation of ABCDX, then XY=CD and CDXY is a regular tetrahedron.

The new polyhedron is formed by ABCDXY. Since AD is parallel to XY, ADXY is a single face. Similarily BCXY is a single face. The other three faces of this solid are ABCD, ABX, and CDY. The new polyhedron has five faces total.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
AnswerK Sengupta2008-11-25 00:06:01
Solutionsolutiondaniel2004-12-18 01:10:18
re(2): Amazing coincidenceKen Haley2004-10-08 00:09:30
re: Amazing coincidencenikki2004-10-06 08:12:03
good onerixar2004-10-06 07:50:23
re(2): Surely not the intended solution ! (spoilers)SilverKnight2004-10-06 03:52:53
Some ThoughtsAmazing coincidenceKen Haley2004-10-06 00:28:20
re: Perhaps simpler calculationsKen Haley2004-10-06 00:20:29
re: Perhaps simpler calculationsDavid Shin2004-10-05 20:36:55
re: Very Simply NeatCharlie2004-10-05 20:25:48
Very Simply NeatCeeAnne2004-10-05 19:15:47
SolutionPerhaps simpler calculationsTristan2004-10-05 18:56:19
Link with FiguresRichard2004-10-05 14:02:51
SolutionAnother way of showing solutionCharlie2004-10-05 13:52:03
re: Surely not the intended solution ! (spoilers)nikki2004-10-05 13:10:14
re(2): Surely not the intended solution !Charlie2004-10-05 13:05:24
re: Surely not the intended solution !Jonathan Fletcher2004-10-05 12:38:22
SolutionSolutionDavid Shin2004-10-05 12:22:56
SolutionSurely not the intended solution !Syzygy2004-10-05 12:08:24
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