Famous Theorems of Mathematics/Geometry/Cones

< Famous Theorems of Mathematics‎ | Geometry

Contents

VolumeEdit

  • Claim: The volume of a conic solid whose base has area b and whose height is h is  .

Proof: Let   be a simple planar loop in  . Let   be the vertex point, outside of the plane of  .

Let the conic solid be parametrized by

 

where  .

For a fixed  , the curve   is planar. Why? Because if   is planar, then since   is just a magnification of  , it is also planar, and   is just a translation of  , so it is planar.

Moreover, the shape of   is similar to the shape of  , and the area enclosed by   is   of the area enclosed by  , which is b.

If the perpendiculars distance from the vertex to the plane of the base is h, then the distance between two slices   and  , separated by   will be  . Thus, the differential volume of a slice is

 

Now integrate the volume:

 

Center of MassEdit

  • Claim: the center of mass of a conic solid lies at one-fourth of the way from the center of mass of the base to the vertex.

Proof: Let   be the total mass of the conic solid where ρ is the uniform density and V is the volume (as given above).

A differential slice enclosed by the curve  , of fixed  , has differential mass

 .

Let us say that the base of the cone has center of mass  . Then the slice at   has center of mass

 .

Thus, the center of mass of the cone should be

 
 
 
 
 
 
 ,

which is to say, that   lies one fourth of the way from   to  .

Dimensional ComparisonEdit

Note that the cone is, in a sense, a higher-dimensional version of a triangle, and that for the case of the triangle, the area is

 

and the centroid lies 1/3 of the way from the center of mass of the base to the vertex.

A tetrahedron is a special type of cone, and it is also a stricter generalization of the triangle.

Surface AreaEdit

  • Claim: The Surface Area of a right circular cone is equal to  , where   is the radius of the cone and   is the slant height equal to  

Proof: The   refers to the area of the base of the cone, which is a circle of radius  . The rest of the formula can be derived as follows.

Cut   slices from the vertex of the cone to points evenly spread along its base. Using a large enough value for   causes these slices to yield a number of triangles, each with a width   and a height  , which is the slant height.

The number of triangles multiplied by   yields  , the circumference of the circle. Integrate the area of each triangle, with respect to its base,  , to obtain the lateral surface area of the cone, A.

 

 

 

 

Thus, the total surface area of the cone is equal to