Conic sections. A cone with water

Pour some water in a trans­parent cone and close it with a lid, which is a base of a cone. Water surface will take a shape of an ellipse, a parabola and a hyper­bola– at different angles of cone. One can also observe some degen­erate cases. In a vertical posi­tion of a cone water surface will be a circle–it is a limiting case of the ellipse. In a hori­zontal posi­tion of a cone water surface will be a triangle– a hyper­bola degen­er­ates into two inter­secting straight lines.

Look at the first image when the cone is posi­tioned verti­cally, pointing down. Which part of the cone do you think is filled with water? Then look at the last image, where the cone is posi­tioned verti­cally, but pointing up. It appears that the almost full cone has become almost empty. However the water volume hasn’t changed, has it? This effect is related to the volume distri­b­u­tion in a cone. In fact, the volume of poured water is half the cone volume. You can see that by looking at the hori­zontal posi­tion of the cone.

Other models in “Conic sections”