This is a model of conic slices that is obtained through reverse-engineering parts of a cone, with its dimensions determined by its isosceles triangular frame and the angle of the cross-sectional slices.

Big Idea

Conic sections

Purpose

To explore conic sections, which are the cross-sections of a cone that include circles, ellipses, parabolas, and hyperbolas

Sample Tasks and Explorations:

1. Using the conic sections model, find the measures of the angles that each conic section makes with the base of the cone.
2. Suppose the base angles of the isosceles triangular frame of a cone measure 30 degrees. What kind of conic section is formed when a cross-section is formed at an angle of:
   90 degrees from the horizontal base?
   60 degrees from the horizontal base?
   30 degrees from the horizontal base?
   15 degrees from the horizontal base?
   0.00000001 degrees from the horizontal base?
3. A circle is formed by cutting a cross section of a cone parallel to its base. Use the model to describe the kinds of cuts that produce an ellipse.
4. What is the difference between the kinds of cuts that produce a circle and the kinds of cuts that produce a hyperbola?
5. What is the difference between the kinds of cuts that produce an ellipse and the kinds of cuts that produce a parabola?