Difference between revisions of "1985 AIME Problems/Problem 15"
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== Problem == | == Problem == | ||
| − | + | Three 12 cm 12 cm [[square (geometry) | squares]] are each cut into two pieces <math>A</math> and <math>B</math>, as shown in the first figure below, by joining the [[midpoint]]s of two adjacent sides. These six pieces are then attached to a [[regular polygon | regular]] [[hexagon]], as shown in the second figure, so as to fold into a [[polyhedron]]. What is the [[volume]] (in <math>\mathrm{cm}^3</math>) of this polyhedron? | |
== Solution == | == Solution == | ||
| − | + | Note that gluing two of the given polyhedra together along a hexagonal face (rotated <math>60^\circ</math> from each other) yields a [[cube (geometry) | cube]], so the volume is <math>\frac12 \cdot 12^3 = 864</math>. | |
== See also == | == See also == | ||
| + | * [[1985 AIME Problems/Problem 14 | Previous problem]] | ||
* [[1985 AIME Problems]] | * [[1985 AIME Problems]] | ||
| + | |||
| + | [[Category:Intermediate Geometry Problems]] | ||
Revision as of 19:23, 1 December 2006
Problem
Three 12 cm 12 cm squares are each cut into two pieces 
and 
, as shown in the first figure below, by joining the midpoints of two adjacent sides. These six pieces are then attached to a regular hexagon, as shown in the second figure, so as to fold into a polyhedron. What is the volume (in 
) of this polyhedron?
Solution
Note that gluing two of the given polyhedra together along a hexagonal face (rotated 
from each other) yields a cube, so the volume is 
.
