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 heptagonal prism is a

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 ...

with

heptagon In geometry, a heptagon or septagon is a seven-sided polygon or 7-gon. The heptagon is sometimes referred to as the septagon, using ''Wikt:septa-, septa-'' (an elision of ''Wikt:septua-, septua-''), a Latin-derived numerical prefix, rather than ...

al base. This

polyhedron In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal Face (geometry), faces, straight Edge (geometry), edges and sharp corners or Vertex (geometry), vertices. The term "polyhedron" may refer ...

has 9 faces (2 bases and 7 sides), 21 edges, and 14 vertices..

Area

The

area Area is the measure of a region's size on a surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open surface or the boundary of a three-di ...

of a right heptagonal prism with height

h

and with a side length of

L

and

apothem The apothem (sometimes abbreviated as apo) of a regular polygon is a line segment from the center to the midpoint of one of its sides. Equivalently, it is the line drawn from the center of the polygon that is perpendicular to one of its sides. T ...

a_p

is given by: :

A = 7L\cdot (a_p + h)

Volume

The

volume Volume is a measure of regions in 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) ...

is found by taking the area of the base, with a side length of

L

and apothem

a_p

, and multiplying it by the height

h

, giving the formula: :

V = \frac\cdot h\cdot L\cdot a_p

This formula also works for the oblique prism due to the

Cavalieri's principle In geometry, Cavalieri's principle, a modern implementation of the method of indivisibles, named after Bonaventura Cavalieri, is as follows: * 2-dimensional case: Suppose two regions in a plane are included between two parallel lines in that pl ...

Images

The heptagonal prism can also be seen as a tiling on a sphere: : Spherical heptagonal prism

Related polyhedra

References

External links

* Prismatoid polyhedra {{Polyhedron-stub