vikasbansal227 wrote:
When the figure above is cut along the solid lines, folded along the dashed lines, and taped along the solid lines, the result is a model of a geometric solid. This geometric solid consists of 2 pyramids each with a square base that they share. What is the sum of number of edges and number of faces of this geometric solid?
A. 10
B. 18
C. 20
D. 24
E. 25
Dear
vikasbansal227I'm happy to respond.
This is a great question.
First of all, here is a labeled version of the polyhedral net you provided:
Attachment:
octahedron net.JPG [ 21.01 KiB | Viewed 71211 times ]
Incidentally, in your version, you indicate a solid line from B to I, but this should be a dashed line.
When this is folded up, the name of the resultant shape is the
Octahedron, one of the five Platonic Solids. It good practice for 3D thinking to study the five Platonic Solids. But, suppose you didn't know about the Octahedron: how would we answer the question?
Notice, that when we fold up, A & C come together, so we could say that B is the vertex of one pyramid, with four edges going down to D, I, J, and A/C. These latter four points form a square that is the "equator" of the shape, so there are four edges around this square. Then, we fold F & H together, forming the second pyramid. This second pyramid has G as its vertex, with four edges come down to the points E, D, I, and F/H; these latter four points also form a square. Then, segment DI acts as a hinge, and along this hinge we fold one pyramid down to meet the other, so that the square bases of the two pyramids meet and become one. Point A/C joins with point E, and point J joins with point F/H.
The final shape has a lower downward-pointing pyramid, with point B as the lower vertex. Four edges go up from this vertex to the square "equator." This square "equator" consists of points D + I + A/C/E + F/H/J, and like any square, it contains four edge segments. Finally, the upper upward-pointing pyramid has vertex G, and four edges go from this vertex down to the four vertices of the square "equator." That's a total of 12 edges.
We can simply count the 8 faces while it is still flat.
E + F = 12 + 8 = 20 ==> OA =
(C) The thing that is a little mind-blowing about the octahedron is this. In the above description, I was discussing the "upper vertex," the "lower vertex," and the "square equator," but because the shape is 100% symmetrical, if we simply turn the shape, any vertex can be the tip of the upper pyramid. There are actually three interlocking squares formed by different combinations of the vertices. This
Wikipedia page has a gif of a rotating octahedron, which may help you visualize it more.
Does all this make sense?
Mike