perplexus.info :: Shapes : A New Solid

A New Solid (Posted on 2004-10-05)

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
Answer	K Sengupta	2008-11-25 00:06:01
solution	daniel	2004-12-18 01:10:18
re(2): Amazing coincidence	Ken Haley	2004-10-08 00:09:30
re: Amazing coincidence	nikki	2004-10-06 08:12:03
good one	rixar	2004-10-06 07:50:23
re(2): Surely not the intended solution ! (spoilers)	SilverKnight	2004-10-06 03:52:53
Amazing coincidence	Ken Haley	2004-10-06 00:28:20
re: Perhaps simpler calculations	Ken Haley	2004-10-06 00:20:29
re: Perhaps simpler calculations	David Shin	2004-10-05 20:36:55
re: Very Simply Neat	Charlie	2004-10-05 20:25:48
Very Simply Neat	CeeAnne	2004-10-05 19:15:47
Perhaps simpler calculations	Tristan	2004-10-05 18:56:19
Link with Figures	Richard	2004-10-05 14:02:51
Another way of showing solution	Charlie	2004-10-05 13:52:03
re: Surely not the intended solution ! (spoilers)	nikki	2004-10-05 13:10:14
re(2): Surely not the intended solution !	Charlie	2004-10-05 13:05:24
re: Surely not the intended solution !	Jonathan Fletcher	2004-10-05 12:38:22
Solution	David Shin	2004-10-05 12:22:56
Surely not the intended solution !	Syzygy	2004-10-05 12:08:24

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