• Skip to main content

Yup Tab

Developing an Equation for a Cone – the Simplest Case

by yup tab

An Idea of a Moment

When I was in high school, I thought it would be fun to derive an equation from scratch for a cone.

A Misconception

Now a cone is not what most people think – a two-dimensional object the shape of an upside down ice cream cone, though it is similar. A cone is really like an upside-down ice cream cone with another ice cream cone that is right side up, with the two tips touching and with a center line running right through the middle of the fat ends. In addition, the cone’s wide ends stretch out both ways to infinity.


We Begin

Now using the basic x-y-z diagram typically used in 3D analytical geometry, I drew a cone (Figure 1) with its cross-over point at the origin, x=y=z=0, or according to accepted notation, (0, 0, 0). Not being able to draw the cone to infinity, I chose to stop the image at y=h at the top and y=-h at the bottom. I define the radius of the cone at y=h as r. Then,

Equation 1. x2 + z2 = r2

Equation 2. r/y = tan (theta) or in other words, r=y tan (theta)

So,

Equation 3. x2 + z2 = y2 tan2 (theta)

Now the angle is a constant in the case of a cone. So this boils down to

Formula: x2 + z2 = ay2

Some Tests

1. Making the constant a equal to zero.

Let’s test this out. If a = 0, then x2 + z2 = 0 Since squared numbers, if real, have to be less than or equal to zero, this is nonsense, unless both x and z are equal to zero. This thus reduces to a line along the y-axis. This makes sense. An infinitesimally thick cone is a line.

2. Making the constant infinite.

If a =∞ , then y could be anything except possibly zero and x2 + z2 = ∞, which would be a cylinder infinitely wide centered along the y-plane. This makes sense.

3. Making z or x or y equal to zero.

If, on the other hand, a was an ordinary number greater than zero and less than infinity, and if z = 0, then x2 = ay2 which can be changed to the following two equations:

x = by where b =√ a

x= -by where b = √a

These two equations constitute two lines crisscrossing in the xy-plane at the origin, (0, 0, 0). This is what a cone becomes when the third dimension, z, is eliminated.

Something similar occurs if x, instead of z, becomes zero, but using the zy-plane without reference to x.

The equations in this instance became:

z = by where b = √a

z = -by where b = √a

If y becomes zero the result is the single point x=y=z=0, or (0,0,0). This is like making the cone have a zero height. This is what we would expect.

In Summary

So our conclusions are correct, and we have the basic equation of a cone with midpoint at the origin and line of symmetry as the y-axis being:

x2 + z2 = ay2

It would have been far more complicated to determine the equation of a cone floating about somewhere in space, with midpoint (x,y,z), and with symmetry not about the x-axis, y-axis, or z-axis. While it would have been difficult, it would have been just as rewarding. No doubt much could be learned from such an undertaking. How about it? Will you try it?

Related

  • Conic Sections: Derived from the Equation for a Cone
  • Ramamia - the Simplest New Way to Privately Share Photos, Videos and Events with Families
  • Five, Fun Pine Cone Craft Ideas for Kids
  • Styrofoam Cone Memorial Bouquet for Headstones
  • Valentine's Day Cupcake Cone Recipe
  • Casey Anthony Case: Case Darkens as 311 Pages of Documents Released

© 2019 Yup Tab · Contact · Privacy