Three Pyramids are Better Than Two

The applet presents a tool for investigating an infamous problem. It showed up in 1980 as a practice problem on the Preliminary Scholastic Aptitude Exam, but, to an embarrassment of the Educational Testing Service, the marked answer was incorrect.

The problem is this [Winkler, p. 43]:

A solid Square Pyramid square-base pyramid, with all edges of unit length, and a solid triangle-base pyramid (tetrahedron), also with all edges of unit length, are glued together by matching two triangular faces. How many faces does the resulting solid have?

(In the applet, objects can be rotated with the left button pressed or translated with the right button pressed. Pressing the left button while the Alt key is down lets one zoom in and away from an object. When the "combine" button is checked the pyramids will move as a single object.)

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at, download and install Java VM and enjoy the applet.

What if applet does not run?

The expected solution was this. A square pyramid has 5 faces and a tetrahedron 4 faces. When two triangular faces are eliminated by gluing them together there remain 5 + 4 - 2 = 7 faces. By adding a second square-base pyramid, it becomes absolutely clear that, in the problem, there are two pairs of adjacent faces that align on a single plane (i.e., are coplanar) so that the resulting solid has two faces less than expected: 5 in all.

(As an aside, what is the relationship between the volumes of the square pyramid and the tertrahedron? Find out.


  1. P. Winkler, Mathematical Puzzles: A Connoisseur's Collection, A K Peters, 2004

Related material

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  • Right Triangular Prism
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  • Tetrahedron: an Interactive Model
  • Octahedron: an Interactive Model
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  • Dodecahedron: an Interactive Model
  • Cube In Octahedron
  • Octahedron In Cube
  • Octahedron In Tetrahedron
  • Tetrahedron In Cube
  • Icosahedron In Cube
  • Great Stellated Dodecahedron
  • Lennes' Polyhedron
  • Triangulated Dinosaur
  • Volumes of Two Pyramids
  • Császár Polyhedron 1
  • Császár Polyhedron 4
  • Szilassi Polyhedron
  • Dissection of a Square Pyramid
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