Study Guide

Conic Sections

Conic Sections Introduction

At a celebration of one of Shmoop's favorite subjects (let’s face it; they’re all our favorite and we just like a celebration), one of our monkey interns got a hold of the cake knife and went all ninja on the party hats, slicing them at various angles. Everyone was upset until someone pointed out that the monkey had just formed four conic sections out of the party hats.

Party hats are shaped like cones, just like ice cream cones. Picture these whenever we’re talking about conic sections. Why? Because we really like parties and ice cream. Also because the conic sections are formed by drawing a line straight through some cones.

A circle is revealed if we slice straight across a cone. If the slice is taken parallel to one of the sloping sides of the cone, then we get a parabola. We would remember this curve from algebra, but we've blocked that out of our memory. It was a hard time for us.

Any cut made with an angle less than the parabola, but more than the circle, will produce an elongated shape called an ellipse. Finally, some slices like the cone so much that they cut it twice, resulting in a hyperbola. They look like two separate curves, but they belong to the same shape.

Everyone immediately recognized the brilliant improvement the monkey had made to the party hats in making them represent one of the greatest geometric marvels in mathematics, conic sections. We all agreed that was one smart monkey.

Conic Sections Resources


Don't let parabolas throw you for a loop. Learn 'em, love 'em, maybe even graph 'em.

Paul's Online Math Notes—Ellipses
On even the most treacherous math problems, Paul always has our back. Tackle ellipses with him, and give him a high-five every time you solve one.

Coolmath—Graphs of Hyperbolas
Looking for a different way to graph hyperbolas? Here's one, and it works pretty well, too.


Cutting Conic Sections From A Cone
Watch in complete silence as the conic sections are cut out of a set of real-life cones. Riveting.

Elicia Team—Ellipses
A 3D rendered film showing different ways ellipse can be drawn. Very classy.

Conics—General Equation and Eccentricity
This guy uses c for the distance from the center to the foci, instead of f like we do. Otherwise, what he says is solid.

Khan Academy—Intro to Hyperbolas
Khaaaan! Academy covers the hyperbolic basics. His neon colors and soothing voice will lead the way.

Games and Tools

Math Playground—Project T.R.I.G.
Test out your parabolic skills with this game demonstrating projectile motion. It’s like Angry Birds with math and less property damage.

Interactive Mathematics—Interactive 3-D Conics Graph
Click and drag around the cross-section of a double-cone, to see what its insides look like. Don't worry, it's not gross, it's a conic section. Well, unless you're grossed out by conic sections. Then it is gross.

Web Graphing—Analyze an Ellipse
Plug in an ellipse equation, get an ellipse graph in return. This is an equivalent exchange we can get behind.