1997 AJHSME Problems/Problem 17
Problem
A cube has eight vertices (corners) and twelve edges. A segment, such as , which joins two vertices not joined by an edge is called a diagonal. Segment
is also a diagonal. How many diagonals does a cube have?
Solution 1
On each face, there are diagonals like
. There are
faces on a cube. Thus, there are
diagonals that are "x-like".
Every "y-like" diagonal must connect the bottom of the cube to the top of the cube. Thus, for each of the bottom vertices of the cube, there is a different "y-like" diagonal. So there are
"y-like" diagonals.
This gives a total of diagonals on the cube, which is answer
.
Solution 2
There are vertices on a cube. If you pick any of these eight points and connect it to any of the other seven points, you will have
segments connecting the eight points. The division by
is necessary because you counted both the segment from
to
and the segment from
to
.
But not all of these segments are diagonals. Some are edges. There are
edges on the top,
edges on the bottom, and
edges that connect the top to the bottom. So there are
edges total, meaning that there are
segments that are not edges. All of these segments are diagonals, and thus the answer is
.
Solution 3
Consider picking one point on the corner of the cube. That point has "x-like" diagonals that end on each of the three planes of the cube, and
"y-like" diagonal that ends on the opposite vertex. Thus, each point has
diagonals associated with it. There are
vertices on the cube, giving a total of
diagonals.
However, each diagonal was counted as both a "starting point" and an "ending point". So there are really diagonals, giving an answer of
.
See also
1997 AJHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 16 |
Followed by Problem 18 | |
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