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### Lateral Surface of a Cone

```
Date: 02/10/99 at 21:08:51
From: Grant
Subject: Lateral surface of cone

What is the equation for the lateral surface of a cone?
```

```
Date: 02/11/99 at 12:51:55
From: Doctor Peterson
Subject: Re: Lateral surface of cone

Hi, Grant. Welcome to the Doctor's office.

I can take your question two ways: the formula for the lateral surface
area, or the equation of the surface itself (involving x, y, and z
coordinates). I'll assume it's the former. You can find the formula on

http://mathforum.org/dr.math/faq/formulas/faq.cone.html

The area is pi*r*s, where r is the radius of the base, and s is the
length of the side (from the vertex to the base). You can figure it
out without too much trouble if you think of the surface as made of a
piece of paper rolled up. Unroll it flat, and it will be a sector of a
circle:

*********
******         ******
***                   /
**                     /
**                      /
*                       /
*                       /s
*                       /
*                       /
*                      /
*                      /
*                     +---------------------*
*                                           *
*                                         *
*                                         *
*                                       *
*                                     *
*                                   *
**                               **
**                           **
***                     *** 2 pi r
******         ******
*********

The arc length of the whole circle would be 2 pi s; but you only have
as much as will fit around the base of the cone, 2 pi r. That means
you have

2 pi r    r
------ = ---
2 pi s    s

of the whole circle, so the area is

r
--- * pi s^2 = pi * r * s
s

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Conic Sections/Circles
High School Geometry
High School Higher-Dimensional Geometry
Middle School Conic Sections/Circles
Middle School Geometry
Middle School Higher-Dimensional Geometry

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