In
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, the augmented hexagonal prism is one of the
Johnson solid
In geometry, a Johnson solid, sometimes also known as a Johnson–Zalgaller solid, is a convex polyhedron whose faces are regular polygons. They are sometimes defined to exclude the uniform polyhedrons. There are ninety-two Solid geometry, s ...
s (). As the name suggests, it can be constructed by augmenting a hexagonal
prism
PRISM is a code name for a program under which the United States National Security Agency (NSA) collects internet communications from various U.S. internet companies. The program is also known by the SIGAD . PRISM collects stored internet ...
by attaching a
square pyramid
In geometry, a square pyramid is a Pyramid (geometry), pyramid with a square base and four triangles, having a total of five faces. If the Apex (geometry), apex of the pyramid is directly above the center of the square, it is a ''right square p ...
() to one of its equatorial faces. When two or three such pyramids are attached, the result may be a
parabiaugmented hexagonal prism (), a
metabiaugmented hexagonal prism (), or a
triaugmented hexagonal prism ().
Construction
The augmented hexagonal prism is constructed by attaching one
equilateral square pyramid
In geometry, a square pyramid is a pyramid with a square base and four triangles, having a total of five faces. If the apex of the pyramid is directly above the center of the square, it is a ''right square pyramid'' with four isosceles triangle ...
onto the square face of a
hexagonal prism
In geometry, the hexagonal prism is a Prism (geometry), prism with hexagonal base. Prisms are polyhedrons; this polyhedron has 8 face (geometry), faces, 18 Edge (geometry), edges, and 12 vertex (geometry), vertices..
As a semiregular polyhedro ...
, a process known as
augmentation. This construction involves the removal of the prism square face and replacing it with the square pyramid, so that there are eleven faces: four
equilateral triangle
An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...
s, five
square
In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...
s, and two regular
hexagon
In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°.
Regular hexagon
A regular hexagon is de ...
s. A convex polyhedron in which all of the faces are regular is a Johnson solid, and the augmented hexagonal prism is among them, enumerated as
. Relatedly, two or three equilateral square pyramids attaching onto more square faces of the prism give more different Johnson solids; these are the
parabiaugmented hexagonal prism , the
metabiaugmented hexagonal prism , and the
triaugmented hexagonal prism .
Properties
An augmented hexagonal prism with edge length
has surface area
the sum of two hexagons, four equilateral triangles, and five squares area. Its volume
can be obtained by slicing into one equilateral square pyramid and one hexagonal prism, and adding their volume up.
It has an
axis of symmetry
An axis (: axes) may refer to:
Mathematics
*A specific line (often a directed line) that plays an important role in some contexts. In particular:
** Coordinate axis of a coordinate system
*** ''x''-axis, ''y''-axis, ''z''-axis, common names f ...
passing through the apex of a square pyramid and the centroid of a prism square face, rotated in a half and full-turn angle. Its
dihedral angle can be obtained by calculating the angle of a square pyramid and a hexagonal prism in the following:
* The dihedral angle of an augmented hexagonal prism between two adjacent triangles is the dihedral angle of an equilateral square pyramid,
* The dihedral angle of an augmented hexagonal prism between two adjacent squares is the interior of a regular hexagon,
* The dihedral angle of an augmented hexagonal prism between square-to-hexagon is the dihedral angle of a hexagonal prism between its base and its lateral face,
* The dihedral angle of a square pyramid between triangle (its lateral face) and square (its base) is
. Therefore, the dihedral angle of an augmented hexagonal prism between square-to-triangle and between triangle-to-hexagon, on the edge in which the square pyramid and hexagonal prism are attached, are
.
References
External links
*
Johnson solids
{{Johnson solids navigator