Question

A regular triangular pyramid (a regular tetrahedron, that is) is sliced by four cutting planes, each of which is parallel to a face of the pyramid and bisects the altitude drawn to that face. This dissects the pyramid into five pieces, four of which are smaller pyramids. Describe the fifth piece, name it, and find its volume.

Answer 1

The fifth piece of the pyramid after it has been sliced by four cutting planes is a triangular prism. The volume of a triangular prism can be calculated using the formula:

Volume = (base area) * (height)

Where base area is the area of the triangular base of the prism and height is the distance from the base to the top of the prism.

In this case, the base of the triangular prism is an equilateral triangle with side length equal to the side length of the original pyramid, and the height of the prism is equal to the altitude of the original pyramid.

So, the volume of the triangular prism can be calculated as follows:

Volume = (base area) * (height)

= [(side length)^2 * sqrt(3)/4] * (height)

= (side length)^2 * sqrt(3)/4 * (height)

Where side length is the side length of the original pyramid and height is the altitude of the original pyramid.

Therefore, to find the volume of the triangular prism, you will need to know the side length and altitude of the original pyramid.