SOLUTION: Calculate the volume of a regular octahedron whose edges are all 10cm

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Question 1106219: Calculate the volume of a regular octahedron whose edges are all 10cm

Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52892)   (Show Source): You can put this solution on YOUR website!
.
From Wikipedia: 

    In geometry, an octahedron (plural: octahedra) is a polyhedron with eight faces, twelve edges, and six vertices. 
    The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, 
    four of which meet at each vertex.


For the solution, see the lesson
    - Solved problems on volume of pyramids
in this site, where the similar problem was solved.



Answer by greenestamps(13209)   (Show Source): You can put this solution on YOUR website!


A regular octahedron is two square pyramids attached base to base.

Each face of the octahedron is an equilateral triangle with side length 10. The altitude of the equilateral triangle (the slant height of the pyramid) has length 5*sqrt(3).

To find the height of the pyramid, consider the right triangle with the height of the pyramid as one leg and the slant height of the pyramid as the hypotenuse. The other leg is half the length of a side of the base. The Pythagorean Theorem gives the height of the pyramid as 5*sqrt(2).

So the volume of the octahedron is the volume of two square pyramids, each with base area 25 and height 5*sqrt(2):
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