Let r = the distance from center to one vertex.
The length (a) of edge, by the Pythagoras Theorem, = r√2.
Then the area of one triangle is (a × h) / 2, where h = √[a² - (a/2)²].
And the area of the octahedron is 8 × the area of one triangle.
|The octahedron can be divided into two pyramids.|
The volume of one pyramid = (base area × height) /3. In the case of the regular octahedron, the base area = a².
And so, the volume of the octahedron = 2 × the volume of pyramid.
V = (√2 / 3)a³