Question 1106219
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A regular octahedron is two square pyramids attached base to base.<br>
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).<br>
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).<br>
So the volume of the octahedron is the volume of two square pyramids, each with base area 25 and height 5*sqrt(2):<br>
{{{V = 2(1/3)(5^2)(5*sqrt(2)) = (250*sqrt(2))/3}}}