Difference between revisions of "2006 AMC 10A Problems/Problem 24"
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== Solution == | == Solution == | ||
− | We can break the octahedron into two [[square pyramid]]s by cutting it along a [[plane]] [[perpendicular]] to one of its internal | + | We can break the octahedron into two [[square pyramid]]s by cutting it along a [[plane]] [[perpendicular]] to one of its internal diagonals. |
<asy> | <asy> | ||
import three; | import three; | ||
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draw((0,r,r)+A--(r,1,r)+A--(1,r,r)+A--(r,0,r)+A--cycle^^(0,r,r)+A--(r,r,1)+A--(1,r,r)+A^^(r,1,r)+A--(r,r,1)+A--(r,0,r)+A); | draw((0,r,r)+A--(r,1,r)+A--(1,r,r)+A--(r,0,r)+A--cycle^^(0,r,r)+A--(r,r,1)+A--(1,r,r)+A^^(r,1,r)+A--(r,r,1)+A--(r,0,r)+A); | ||
</asy> | </asy> | ||
− | The cube has | + | The cube has edges of length 1 so all edges of the regular octahedron have length <math>\frac{\sqrt{2}}{2}</math>. Then the square base of the [[pyramid]] has area <math>\left(\frac{1}{2}\sqrt{2}\right)^2 = \frac{1}{2}</math>. |
− | We also know that the height of the pyramid is half the height of the cube, so it is <math>\frac{1}{2}</math>. The volume of a pyramid with base area <math>B</math> and height <math>h</math> is <math>A=\frac{1}{3}Bh</math> so each of the pyramids has volume <math>\frac{1}{3}\left(\frac{1}{2}\right)\left(\frac{1}{2}\right) = \frac{1}{12}</math>. The whole octahedron is twice this volume, so <math>\frac{1}{12} \cdot 2 = \frac{1}{6} | + | We also know that the height of the pyramid is half the height of the cube, so it is <math>\frac{1}{2}</math>. The volume of a pyramid with base area <math>B</math> and height <math>h</math> is <math>A=\frac{1}{3}Bh</math> so each of the pyramids has volume <math>\frac{1}{3}\left(\frac{1}{2}\right)\left(\frac{1}{2}\right) = \frac{1}{12}</math>. The whole octahedron is twice this volume, so <math>\frac{1}{12} \cdot 2 = \boxed{\textbf{(B) }\frac{1}{6}}</math>. |
== See also == | == See also == |
Latest revision as of 10:19, 19 December 2021
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 square pyramids by cutting it along a plane perpendicular to one of its internal diagonals. The cube has edges of length 1 so all edges of the regular octahedron have length . Then the square base of the pyramid has area . We also know that the height of the pyramid is half the height of the cube, so it is . The volume of a pyramid with base area and height is so each of the pyramids has volume . 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 |
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