Visualize two identical right circular cones such that one is inverted over the other and they share a common circular base. If a cutting plane passes through the vertices of the assembled cones, what shape does the outer boundary of the resulting cross-section make?
Answer
(A)
Explanation
Joined base to base, the two cones share one axis, which passes through both vertices and the centre of the common base. A plane through both vertices therefore contains this axis and cuts each cone in an isosceles triangle: its base is a diameter of the circle and its equal sides are slant heights.
The two triangles sit on opposite sides of the same diameter, so the outer boundary is a quadrilateral whose four sides are all equal slant heights: a rhombus. It cannot be an ellipse, because a plane through the axis meets a cone’s curved surface in straight lines.
Answer: A