Since the formula for the lateral area of a right cone is the same as the formula for the area of an ellipse, are there any deeper connections between these two geometric objects, other than the fact that an ellipse is a conic section?
- Anybody can ask a question
- Anybody can answer
- The best answers are voted up and rise to the top
Here is an intuitive explanation of why they should be equal:
Imagine taking a cone of radius $r$ and slant height $l$, removing the base of the cone, and putting it in a vise (the sides of the vise will be exactly $2r$ apart). Now squash the point of the cone down flat - the sides of the vise are preventing any expansion in one direction, so all of the extra "material" of the cone goes into extending it's footprint along the axis parallel to the sides of the vise. The result will be an ellipse with one radius equal to $r$ (this is perpendicular to the sides of the vise), and the other radius equal to the slant height (this is parallel to the sides of the vise).
So, the area of the "lateral area" part of a cone (i.e. the surface area, minus the base) is equal to the area of an ellipse with radii equal to $r$ and $l$.