Take a rectangular prism. Choose a point P along a vertical rod through the center of the base. Draw lines from P to the four vertices of the prism base so that P is the apex of a rectangular pyramid.
A GSP drawing. Move P gently.
Question: How should the point P be selected so that the volume of the "base pyramid" equals the volume of a pyramid drawn using one of the sides as the base?
Label the figure and draw PM, the altitude from P to the lateral side (i.e. the line segment perpendicular to the plane of the side). Let k be the distance along the rod from the base to P (i.e., the altitude of the pyramid.
The volume of a pyramid is one third the area of the base times the altitude. Write equations for the volumes of the two pyramids to compare. What is the altitude of the pyramid with a lateral side of the prisim as its base? Does this altitude length depend on the location of point P on the rod? Why?
Question: Does it matter which of the sides is selected? That is, does the location of P depend on which of the four lateral sides is selected?
Obviously, if the top of the prism is selected for the second base, the location of P to give equal volumes to the two pyramids is the midpoint of the rod.