Volume

• 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 λ = λ₀, 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 α(t), and the area enclosed by is of the area enclosed by , which is b.

If the perpendicular distance from the vertex to the plane of the base is h, then the distance between two slices λ = λ₀ and λ = λ₁, separated by dλ = λ₁ − λ₀ will be . Thus, the differential volume of a slice is

Now integrate the volume:

Q.E.D. Q.E.D. is an abbreviation of the Latin phrase (literally, which was to be demonstrated). In simple terms, the use of this Latin phrase is to indicate that something has been definitively proven. ...

Center of mass

• 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 M = ρV be the total mass of the conic solid where ρ is the uniform density and V is the volume (as given above). In physics, the center of mass of a system of particles is a specific point at which, for many purposes, the systems mass behaves as if it were concentrated. ...

A differential slice enclosed by the curve , of fixed λ = λ₀, has differential mass

.

Let us say that the base of the cone has center of mass . Then the slice at λ = λ₀ 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 , Q.E.D. Q.E.D. is an abbreviation of the Latin phrase (literally, which was to be demonstrated). In simple terms, the use of this Latin phrase is to indicate that something has been definitively proven. ...

Dimensional comparison

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 A triangle is one of the basic shapes of geometry: a polygon with three vertices and three sides which are straight line segments. ...

and the centroid lies 1/3 of the way from the center of mass of the base to the vertex. Centroid of a triangle In geometry, the centroid or barycenter of an object in -dimensional space is the intersection of all hyperplanes that divide into two parts of equal moment about the hyperplane. ...

A tetrahedron is a special type of cone, and it is also a stricter generalization of the triangle. A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ...