   SEARCH HOME Math Central Quandaries & Queries  Question from Jean, a student: "Two great circles lying in planes that are perpendicular to each other are drawn on a wooden sphere of radius "a". Part of the sphere is then shaved off in such a way that each cross section of the remaining solid that is perpendicular to the common diameter of the two great circles is a square whose vertices lie on these circles. Find the volume of this solid." I don't understand the geometry of the problem. Can you please explain the problem and if possible draw a diagram for me? It's easy if you know how to integrate. First, for a picture of half the object, look at our web page:
mathcentral.uregina.ca/QQ/database/QQ.09.06/danielle1.html

This shape can be used for domes of public buildings, almost like the Saskatchewan Legislative Building,

where I say "almost" because they used perpendicular ellipses rather than circles for the dome.

For the volume I would I would take the common diameter to be the x-axis -- just rotate the first figure clockwise so that one of the circles becomes the equator. I recommend that you find the volume of the right half, then multiply that number by 2 for your final answer. Thus you integrate the area of the square cross section times dx, from x = 0 to x = a. For each x you know the diagonal of the square to be y = 2*sqrt(a^2 - x^2). From that you can get the side of the square and then the area of the cross section.

Chris     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.