Tuesday, January 12, 2010

PROJECT ON CONE



LOGICAL DIAGRAM OF CONE :

Cone

A cone is a surface generated by a family of all lines through a given point (the vertex)
and passing through a curve in a plane (the directrix). More commonly, a cone includes
the solid enclosed by a cone and the plane of the directrix. The region of the plane
enclosed by the directrix is called a base of the cone. The perpendicular distance from
the vertex to the plane of the base is the height of the cone.


Height: h
Area of base: B
Volume: V
V = hB/3
Circular Cone


A cone whose base is a circle. The line connecting the center of the base to the vertex is called the axis of the circular cone.

Right Circular Cone

 In a right circular cone, the axis is perpendicular to the base. (If the axis of a
circular cone is not perpendicular to the base, it is called an oblique circular cone.)
The length of any line segment connecting the vertex to the directrix is called the
slant height of the cone.

Height: h
Radius of base: r
Slant height: s
Lateral surface area: S
Total surface area: T
Volume: V

B = Pi r2
s = sqrt[r2+h2]
S = Pi rs
T = Pi r(r+s)
V = Pi r2h/3



(Learn how to build a cone from paper or other flat material.)

Frustum of a Right Circular Cone

The part of a right circular cone between the base and a plane parallel to the base
whose distance from the base is less than the height of the cone.


Height: h
Radius of bases: r, R
Slant height: s
Lateral surface area: S
Total surface area: T
Volume: V

s = sqrt([R-r]2+h2)
S = Pi(r+R)s
T = Pi(r[r+s]+R[R+s])
V = Pi(R2+rR+r2)h/3

(Learn how to build a frustum from paper or other flat material.)


VOLUME of cone = (1/3) b h = 1/3 [pi] r2 h

CONES: The rules are simple.

  1. The layout radius is the length of the side of the cone. For truncated cones the lengths are to the theoretical point (c1 + c2).


  2. The FLAT layout angle is determined by the circumference of the finished cone divided by the circumference of the layout circle (from radius 1. above).


  3. Since PI cancels in the two circumferences above you divide the radius of the base (b2) by the length of the side (for truncated cones use the length to the theoretical point c1 + c2). This is the ratio or percentage of the layout circle. This is also the sine of half the angle of the top of the cone (angle B).

So, if you want a cone with a 90° point then the ratio is the sine of half the angle of the point (B above).

sine(45°) = .7071

Therefore the layout angle S = .7071 x 360° = 254.6°

You can use either method, the ratio or the sine of the angle. Both numbers are the same. sine is simply a word for ratio in geometry. If you have a starting angle it is easy to use a modern calculator to determine the sine.


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
this page:

  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





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