Question

Alice is making an octahedron out of origami. The last step requires her to blow into the octahedron to fill it up. A square pyramid on top of an upside down square pyramid the same size. The length of each side of the square base is a and the height of each pyramid is b. Which expression represents the volume of air she will need to inflate the octahedron?

Answer 1

Total volume is volumes of both square pyramids.

Volume of a tetrahedron is 1/3 times corresponding prism, and a square pyramid has two tetrahedrons.

Thus, volume of on square pyramid:

$= (\frac{1}{3}\times \frac{a^2\times b}{2})\times 2\\=\frac{1}{3}a^2b$

Since there are two of them, total volume is:

$\boxed{\frac{2a^2b}{3}}$