Difference between revisions of "2006 AMC 10A Problems/Problem 24"
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== Problem == | == Problem == | ||
− | + | [[Center]]s of adjacent faces of a unit [[cube (geometry) | cube]] are joined to form a regular [[octahedron]]. What is the [[volume]] of this octahedron? | |
<math>\mathrm{(A) \ } \frac{1}{8}\qquad\mathrm{(B) \ } \frac{1}{6}\qquad\mathrm{(C) \ } \frac{1}{4}\qquad\mathrm{(D) \ } \frac{1}{3}\qquad\mathrm{(E) \ } \frac{1}{2}\qquad</math> | <math>\mathrm{(A) \ } \frac{1}{8}\qquad\mathrm{(B) \ } \frac{1}{6}\qquad\mathrm{(C) \ } \frac{1}{4}\qquad\mathrm{(D) \ } \frac{1}{3}\qquad\mathrm{(E) \ } \frac{1}{2}\qquad</math> | ||
== Solution == | == Solution == | ||
− | {{ | + | We can break the octahedron into two [[tetrahedron]]s. The cube has sides of length one, so we can solve for the area of the tetrahedron's base: <math>(\frac{1}{2}\sqrt{2})^2 = \frac{1}{2}</math>. |
− | + | We also know that the height of a tetrahedron, which would be half the height of the cube, is <math>\frac{1}{2}</math>. Using the volume formula of a tetrahedron: <math>A=\frac{1}{3}Bh</math> where B is the base side length, we find that the volume of one of the tetrahedrons is <math>\frac{1}{3}(\frac{1}{2})(\frac{1}{2}) = \frac{1}{12}</math>. The whole octahedron is twice this volume, so <math>\frac{1}{12} \cdot 2 = \frac{1}{6} \Rightarrow \mathrm{B}</math>. | |
− | + | == See also == | |
− | + | {{AMC10 box|year=2006|ab=A|num-b=23|num-a=25}} | |
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[[Category:Introductory Geometry Problems]] | [[Category:Introductory Geometry Problems]] |
Revision as of 08:27, 15 February 2007
Problem
Centers of adjacent faces of a unit cube are joined to form a regular octahedron. What is the volume of this octahedron?
Solution
We can break the octahedron into two tetrahedrons. The cube has sides of length one, so we can solve for the area of the tetrahedron's base: . We also know that the height of a tetrahedron, which would be half the height of the cube, is . Using the volume formula of a tetrahedron: where B is the base side length, we find that the volume of one of the tetrahedrons is . The whole octahedron is twice this volume, so .
See also
2006 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 23 |
Followed by Problem 25 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |