Frustum
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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 (): :V = \frac\left(a^2 + ab + b^2\right), 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": :V = \frac, 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 :\frac = \frac = \frac = \alpha, the formula for the volume can be expressed as the third of the product of this proportionality, \alpha, and of the difference of the cubes of the heights and only: :V = \frac = \alpha\frac. By using the identity , one gets: :V = (h_1 - h_2)\alpha\frac, where is the height of the frustum. Distributing \alpha and substituting from its definition, the Heronian mean of areas and is obtained: :\frac; the alternative formula is therefore: :V = \frac\left(B_1 + \sqrt + B_2\right). 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: ::V = \frac\left(r_1^2 + r_1r_2 + r_2^2\right), :where and are the base and top radii. *The volume of a pyramidal frustum whose bases are regular -gons is: ::V = \frac\left(a_1^2 + a_1a_2 + a_2^2\right)\cot\frac, :where and are the base and top side lengths. :


Surface area

For a right circular conical frustum :\begin\text&=\pi\left(r_1+r_2\right)s\\ &=\pi\left(r_1+r_2\right)\sqrt\end and :\begin\text&=\pi\left(\left(r_1+r_2\right)s+r_1^2+r_2^2\right)\\ &=\pi\left(\left(r_1+r_2\right)\sqrt+r_1^2+r_2^2\right)\end 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 :A= \frac\left left(a_1^2+a_2^2\right)\cot \frac + \sqrt \right/math> where ''a''1 and ''a''2 are the sides of the two bases.


Examples

*On the back (the reverse) of a
United States one-dollar bill The United States one-dollar bill ($1), sometimes referred to as a single, has been the lowest value denomination of United States paper currency since the discontinuation of U.S. fractional currency notes in 1876. An image of the first U. ...
, a pyramidal frustum appears on the reverse of the Great Seal of the United States, surmounted by the Eye of Providence. *
Ziggurat A ziggurat (; Cuneiform: 𒅆𒂍𒉪, Akkadian: ', D-stem of ' 'to protrude, to build high', cognate with other Semitic languages like Hebrew ''zaqar'' (זָקַר) 'protrude') is a type of massive structure built in ancient Mesopotamia. It has ...
s, step pyramids, and certain ancient Native American mounds also form the frustum of one or more pyramids, with additional features such as stairs added. * Chinese pyramids. *The
John Hancock Center The John Hancock Center is a 100- story, 1,128-foot supertall skyscraper located in Chicago, Illinois. Located in the Magnificent Mile district, the building was officially renamed 875 North Michigan Avenue in 2018. The skyscraper was designe ...
in
Chicago (''City in a Garden''); I Will , image_map = , map_caption = Interactive Map of Chicago , coordinates = , coordinates_footnotes = , subdivision_type = List of sovereign states, Count ...
,
Illinois Illinois ( ) is a state in the Midwestern United States. Its largest metropolitan areas include the Chicago metropolitan area, and the Metro East section, of Greater St. Louis. Other smaller metropolitan areas include, Peoria and Roc ...
is a frustum whose bases are rectangles. *The
Washington Monument The Washington Monument is an obelisk shaped building within the National Mall in Washington, D.C., built to commemorate George Washington, once commander-in-chief of the Continental Army (1775–1784) in the American Revolutionary War and ...
is a narrow square-based pyramidal frustum topped by a small pyramid. *The viewing frustum in
3D computer graphics 3D computer graphics, or “3D graphics,” sometimes called CGI, 3D-CGI or three-dimensional computer graphics are graphics that use a three-dimensional representation of geometric data (often Cartesian) that is stored in the computer for t ...
is a virtual photographic or video camera's usable field of view modeled as a pyramidal frustum. *In the English translation of Stanislaw Lem's short-story collection '' The Cyberiad'', the poem ''Love and tensor algebra'' claims that "every frustum longs to be a cone". * Buckets and typical
lampshade A lampshade is a fixture that envelops the lightbulb on a lamp to diffuse the light it emits. Lampshades can be made out of a large variety of materials like paper, glass, fabric or stone. Often times conical or cylindrical in shape, lampsh ...
s are everyday examples of conical frustums. *Drinking glasses and some space capsules are also some examples. * Garsų Gaudyklė wooden structure or statue in Lithuania.


See also

*
Spherical frustum In geometry, a spherical segment is the solid defined by cutting a sphere or a ball with a pair of parallel planes. It can be thought of as a spherical cap with the top truncated, and so it corresponds to a spherical frustum. The surface of th ...


Notes


References


External links


Derivation of formula for the volume of frustums of pyramid and cone
(Mathalino.com) * *
Paper models of frustums (truncated pyramids)Paper model of frustum (truncated cone)Design paper models of conical frustum (truncated cones)
{{Authority control Polyhedra Prismatoid polyhedra