Frustum

# Frustum

Overview
In geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, a frustum (plural: frusta or frustums) is the portion of a solid
Polyhedron
In elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...

(normally a cone
Cone (geometry)
A cone is an n-dimensional geometric shape that tapers smoothly from a base to a point called the apex or vertex. Formally, it is the solid figure formed by the locus of all straight line segments that join the apex to the base...

or pyramid
Pyramid (geometry)
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. It is a conic solid with polygonal base....

) that lies between two parallel planes cutting it.
Discussion

Encyclopedia
In geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, a frustum (plural: frusta or frustums) is the portion of a solid
Polyhedron
In elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...

(normally a cone
Cone (geometry)
A cone is an n-dimensional geometric shape that tapers smoothly from a base to a point called the apex or vertex. Formally, it is the solid figure formed by the locus of all straight line segments that join the apex to the base...

or pyramid
Pyramid (geometry)
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. It is a conic solid with polygonal base....

) that lies between two parallel planes cutting it.

The term is commonly used in computer graphics
Computer graphics
Computer graphics are graphics created using computers and, more generally, the representation and manipulation of image data by a computer with help from specialized software and hardware....

to describe the three-dimensional region which is visible on the screen, the 'viewing frustum
Viewing frustum
In 3D computer graphics, the viewing frustum or view frustum is the region of space in the modeled world that may appear on the screen; it is the field of view of the notional camera. The exact shape of this region varies depending on what kind of camera lens is being simulated, but typically it is...

', which is formed by a clipped
Clipping (computer graphics)
Any procedure which identifies that portion of a picture which is either inside or outside a picture is referred to as a clipping algorithm or clipping.The region against which an object is to be clipped is called clipping window.-Examples:...

pyramid; in particular, frustum culling is a method of hidden surface determination
Hidden surface determination
In 3D computer graphics, hidden surface determination is the process used to determine which surfaces and parts of surfaces are not visible from a certain viewpoint...

.

In the aerospace industry, frustum is the common term for the fairing
Payload fairing is one of the main components of a launch vehicle. The fairing protects the payload during the ascent against the impact of the atmosphere . More recently, an additional function is to maintain the cleanroom environment for precision instruments.Outside the atmosphere the fairing is...

between two stages of a multistage rocket
Multistage rocket
A multistage rocket is a rocket that usestwo or more stages, each of which contains its own engines and propellant. A tandem or serial stage is mounted on top of another stage; a parallel stage is attached alongside another stage. The result is effectively two or more rockets stacked on top of or...

(such as the Saturn V
Saturn V
The Saturn V was an American human-rated expendable rocket used by NASA's Apollo and Skylab programs from 1967 until 1973. A multistage liquid-fueled launch vehicle, NASA launched 13 Saturn Vs from the Kennedy Space Center, Florida with no loss of crew or payload...

), which is shaped like a truncated
Truncation (geometry)
In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new facet in place of each vertex.- Uniform truncation :...

cone.

## Elements, special cases, and related concepts

Each plane section is a floor or base of the frustum. Its axis if any, 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
Apex (geometry)
In geometry, an apex is the vertex which is in some sense the highest of the figure to which it belongs.*In an isosceles triangle, the apex is the vertex where the two sides of equal length meet, opposite the unequal third side....

(so that the corresponding base reduces to a point). The pyramidal frusta are a subclass of the prismatoid
Prismatoid
In geometry, a prismatoid is a polyhedron where all vertices lie in two parallel planes....

s.

Two frusta joined at their bases make a bifrustum
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....

.

### Volume

The volume
Volume
Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....

of a conical or pyramidal frustum is the volume of the solid before slicing the apex off, minus the volume of the apex:
where B1 is the area of one base, B2 is the area of the other base, and h1, h2 are the perpendicular heights from the apex to the planes of the two bases.

Considering that
the volume can also be expressed as the product of the height h = h2h1 of the frustum, and the Heronian mean of their areas:
Heron of Alexandria is noted for deriving this formula and with it encountering the imaginary number
Imaginary number
An imaginary number is any number whose square is a real number less than zero. When any real number is squared, the result is never negative, but the square of an imaginary number is always negative...

, the square root of negative one.

In particular, the volume of a circular cone frustum is
where π
Pi
' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...

is 3.14159265..., and R1, R2 are the radii of the two bases.

The volume of a pyramidal frustum whose bases are n-sided regular polygons is
where a1 and a2 are the sides of the two bases.

### Surface area

For a right circular conical frustum
and
where R1 and R2 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 polygons
Polygon
In geometry a polygon is a flat shape consisting of straight lines that are joined to form a closed chain orcircuit.A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments...

is
where a1 and a2 are the sides of the two bases.

## Examples

• A notable example of a pyramidal frustum appears on the reverse of the Great Seal of the United States
Great Seal of the United States
The Great Seal of the United States is used to authenticate certain documents issued by the United States federal government. The phrase is used both for the physical seal itself , and more generally for the design impressed upon it...

on the back of the United States one-dollar bill
United States one-dollar bill
The United States one-dollar bill is the most common denomination of US currency. The first president, George Washington, painted by Gilbert Stuart, is currently featured on the obverse, while the Great Seal of the United States is featured on the reverse. The one-dollar bill has the oldest...

, the "unfinished pyramid" being surmounted by the Eye of Providence
Eye of Providence
The Eye of Providence is a symbol showing an eye often surrounded by rays of light or a glory and usually enclosed by a triangle...

.
• Certain ancient Native American
Indigenous peoples of the Americas
The indigenous peoples of the Americas are the pre-Columbian inhabitants of North and South America, their descendants and other ethnic groups who are identified with those peoples. Indigenous peoples are known in Canada as Aboriginal peoples, and in the United States as Native Americans...

mounds also form the frustum of a pyramid.
• Chinese pyramids
Chinese pyramids
Chinese pyramids are ancient mausoleums and burial mounds built to house the remains of several early emperors of China and their imperial relatives. About 38 of them are located around - north-west of Xi'an, on the Qin Chuan Plains in Shaanxi Province...

• The John Hancock Center
John Hancock Center
John Hancock Center at 875 North Michigan Avenue in the Streeterville area of Chicago, Illinois, is a 100-story, 1,127-foot tall skyscraper, constructed under the supervision of Skidmore, Owings and Merrill, with chief designer Bruce Graham and structural engineer Fazlur Khan...

in Chicago
Chicago
Chicago is the largest city in the US state of Illinois. With nearly 2.7 million residents, it is the most populous city in the Midwestern United States and the third most populous in the US, after New York City and Los Angeles...

, Illinois
Illinois
Illinois is the fifth-most populous state of the United States of America, and is often noted for being a microcosm of the entire country. With Chicago in the northeast, small industrial cities and great agricultural productivity in central and northern Illinois, and natural resources like coal,...

is a frustum whose bases are rectangles.
• The Washington Monument
Washington Monument
The Washington Monument is an obelisk near the west end of the National Mall in Washington, D.C., built to commemorate the first U.S. president, General George Washington...

is a narrow pyramidal frustum (with square bases) with a pyramid attached to the top base.
• The viewing frustum
Viewing frustum
In 3D computer graphics, the viewing frustum or view frustum is the region of space in the modeled world that may appear on the screen; it is the field of view of the notional camera. The exact shape of this region varies depending on what kind of camera lens is being simulated, but typically it is...

in 3D computer graphics
3D computer graphics
3D computer graphics are graphics that use a three-dimensional representation of geometric data that is stored in the computer for the purposes of performing calculations and rendering 2D images...

is the usable field of view of a virtual photographic or video camera
Camera
A camera is a device that records and stores images. These images may be still photographs or moving images such as videos or movies. The term camera comes from the camera obscura , an early mechanism for projecting images...

modeled as a pyramidal frustum.
• The poem Love and tensor algebra, in the English
English language
English is a West Germanic language that arose in the Anglo-Saxon kingdoms of England and spread into what was to become south-east Scotland under the influence of the Anglian medieval kingdom of Northumbria...

translation of Stanislaw Lem
Stanislaw Lem
Stanisław Lem was a Polish writer of science fiction, philosophy and satire. He was named a Knight of the Order of the White Eagle. His books have been translated into 41 languages and have sold over 27 million copies. He is perhaps best known as the author of the 1961 novel Solaris, which has...