Famous Theorems of Mathematics/Geometry/Cones

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  • 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 Mass[edit]

  • 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 Comparison[edit]

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 Area[edit]

  • 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