Find the volume of a pyramid inscribed in a cube if the edge of the cube is 3.

The volume of the pyramid is determined by the formula:

V = 1/3 * S * h,

where V is the volume of the pyramid,
S is the area of ​​the base of the pyramid,
h is the height of the pyramid.

Since the pyramid is inscribed in a cube, and the cube is a regular polyhedron, each face of which is a square, it can be argued that the height of the pyramid is equal to the edge of the cube. That is: h = 3.

In addition, the area of ​​the base of the pyramid is equal to the area of ​​the cube face S = Scube, since the base of the pyramid is located on one of the cube faces (see figure). In turn, the area of ​​the cube face is equal to the edge of the cube squared Scube = a2 (a is the side of the cube face).

As a result, we get the following formula for the volume of a pyramid inscribed in a cube:

V = 1/3 * S * h = 1/3 * a ^ 2 * a.

Substitute the values ​​and determine the volume of the pyramid:

V = 1/3 * 3 ^ 2 * 3 = 3 ^ 2 = 9.

Answer: the volume of a pyramid inscribed in a cube is 9 units.