truncated cone

Think a general cone (not necessarily a circular one). If a plane intersects the lateral surface of the cone, but not the base of this cone, then the solid remaining from the cone between the intersecting plane and the plane of the base A is called a truncated cone. Also the name frustum (the Latin frustum = ‘piece, fragment’) is used, althouh this may mean the portion of any solid which lies between two parallel planesMathworldPlanetmath cutting the solid. Sometimes one sees the (wrong) variant name frustrum. As the plural form of frustum both frustums and frusta are used.

We restrict our to the frustum of cone with the cutting plane parallelMathworldPlanetmath to the plane of the base. The part of its surface contained in the intersecting plane is the other base A of the frustum. Since the bases A and A are homotheticMathworldPlanetmath with respect to the apex of the whole cone, they are similarMathworldPlanetmath planar figures. The height h of the frustum is the part of the height H of the whole cone between the both base planes. Denote  h:=H-h.

The volume of the frustum is obtained as the difference of the volumes of two cones:

V=13AH-13Ah=13[Ah+(A-A)h] (1)

One needs the term (A-A)h.  The ratio of the similar areas A and A equals the square of the line ratio:


Thus we have the proportion equation


subtracting 1 from both sides, it may be written






Considering this in (1) we arrive to the volume formula of the frustum:

V=A+AA+A3h (2)

The quotient in front of h is called the Heronian mean of the positive numbers A and A.


  • 1 K. Väisälä: Geometria.  Reprint of the tenth edition.  Werner Söderström Osakeyhtiö, Porvoo & Helsinki (1971).
Title truncated cone
Canonical name TruncatedCone
Date of creation 2013-03-22 17:48:01
Last modified on 2013-03-22 17:48:01
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 11
Author pahio (2872)
Entry type Derivation
Classification msc 51M04
Classification msc 01A20
Classification msc 51M20
Synonym frustum
Synonym frusta
Synonym frustrum
Related topic PrismatoidMathworldPlanetmath
Related topic KalleVaisala
Defines height of frustum
Defines bases of frustum
Defines Heronian mean