In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a (from the Latin for "morsel"; plural: ''frusta'' or ''frustums'') is the portion of a
solid
Solid is one of the four fundamental states of matter (the others being liquid, gas, and plasma). The molecules in a solid are closely packed together and contain the least amount of kinetic energy. A solid is characterized by structura ...
(normally a
pyramid
A pyramid (from el, πυραμίς ') is a structure whose outer surfaces are triangular and converge to a single step at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrila ...
or a
cone) that lies between two
parallel planes cutting this solid. In the case of a pyramid, the base faces are
polygonal, the side faces are
trapezoidal. A right frustum is a
right pyramid
In geometry, a pyramid () is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle, called a ''lateral face''. It is a conic solid with polygonal base. A pyramid with an base ...
or a right cone
truncated perpendicularly to its axis; otherwise it is an oblique frustum.
If all its edges are forced to become of the same length, then a frustum becomes a
prism (possibly oblique or/and with irregular bases).
In
computer graphics
Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great deal ...
, the
viewing frustum is the three-dimensional region which is visible on the screen. It is formed by a
clipped
''Clipped'' is a video featuring five tracks by the Australian hard rock band AC/DC. First released in 1991, it contained three tracks from '' The Razors Edge'' and two from ''Blow Up Your Video''.
In 2002 a DVD version was released which al ...
pyramid; in particular, ''
frustum culling'' is a method of
hidden surface determination.
In the
aerospace industry, a frustum is the
fairing between two stages of a
multistage rocket (such as the
Saturn V
Saturn V is a retired American super heavy-lift launch vehicle developed by NASA under the Apollo program for human exploration of the Moon. The rocket was human-rated, with three stages, and powered with liquid fuel. It was flown from 1 ...
), which is shaped like a
truncated cone.
Elements, special cases, and related concepts

A frustum's axis is that of the original cone or pyramid. A frustum is circular if it has circular bases; it is right if the axis is perpendicular to both bases, and oblique otherwise.
The height of a frustum is the perpendicular distance between the planes of the two bases.
Cones and pyramids can be viewed as degenerate cases of frusta, where one of the cutting planes passes through the
apex (so that the corresponding base reduces to a point). The pyramidal frusta are a subclass of
prismatoids.
Two frusta with two
congruent bases joined at these congruent bases make a
bifrustum
An ''n''-agonal bifrustum is a polyhedron composed of three parallel planes of ''n''-agons, with the middle plane largest and usually the top and bottom congruent.
It can be constructed as two congruent frusta combined across a plane of symmetry ...
.
Formulas
Volume
The formula for the volume of a pyramidal square frustum was introduced by the ancient
Egyptian mathematics in what is called the
Moscow Mathematical Papyrus, written in the
13th dynasty ():
:
where and are the base and top side lengths, and is the height.
The Egyptians knew the correct formula for the volume of such a truncated square pyramid, but no proof of this equation is given in the Moscow papyrus.
The
volume
Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). Th ...
of a conical or pyramidal frustum is the volume of the solid before slicing its "apex" off, minus the volume of this "apex":
:
where and are the base and top areas, and and are the perpendicular heights from the apex to the base and top planes.
Considering that
:
the formula for the volume can be expressed as the third of the product of this proportionality,
, and of the
difference of the cubes of the heights and only:
:
By using the identity , one gets:
:
where is the height of the frustum.
Distributing
and substituting from its definition, the
Heronian mean of areas and is obtained:
:
the alternative formula is therefore:
:
Heron of Alexandria is noted for deriving this formula, and with it, encountering the
imaginary unit
The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition a ...
: the square root of negative one.
In particular:
*The volume of a circular cone frustum is:
::
:where and are the base and top
radii.
*The volume of a pyramidal frustum whose bases are regular -gons is:
::
:where and are the base and top side lengths.
:
Surface area

For a right circular conical frustum
:
and
:
where ''r''
1 and ''r''
2 are the base and top radii respectively, and ''s'' is the slant height of the frustum.
The surface area of a right frustum whose bases are similar regular ''n''-sided
polygon
In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed '' polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two t ...
s is
: