Editing Polyhedron (section)

===Volume===
Polyhedral solids have an associated quantity called [[volume]] that measures how much space they occupy.  Simple families of solids may have simple formulas for their volumes; for example, the volumes of pyramids, prisms, and [[parallelepiped]]s can easily be expressed in terms of their edge lengths or other coordinates.  (See [[volume#Formulas|Volume § Volume formulas]] for a list that includes many of these formulas.)

Volumes of more complicated polyhedra may not have simple formulas. The volumes of such polyhedra may be computed by subdividing the polyhedron into smaller pieces (for example, by [[point-set triangulation|triangulation]]).  For example, the [[Platonic solid#Radii, area, and volume|volume of a Platonic solid]] can be computed by dividing it into congruent [[Pyramid (geometry)|pyramids]], with each pyramid having a face of the polyhedron as its base and the centre of the polyhedron as its apex.

In general, it can be derived from the [[divergence theorem]] that the volume of a polyhedral solid is given by
<math display="block">
\frac{1}{3} \left| \sum_F (Q_F \cdot N_F) \operatorname{area}(F) \right|,
</math>
where the sum is over faces <math> F </math> of the polyhedron, <math> Q_F </math> is an arbitrary point on face <math> F </math>, <math> N_F </math> is the [[unit vector]] perpendicular to <math> F </math> pointing outside the solid, and the multiplication dot is the [[dot product]].<ref>{{citation |last=Goldman |first=Ronald N.|author-link=Ron Goldman (mathematician) |editor-last=Arvo |editor-first=James |title=Graphic Gems Package: Graphics Gems II |publisher=Academic Press |year=1991 |pages=170–171 |chapter=Chapter IV.1: Area of planar polygons and volume of polyhedra}}</ref> In higher dimensions, volume computation may be challenging, in part because of the difficulty of listing the faces of a convex polyhedron specified only by its vertices, and there exist specialized [[algorithm]]s to determine the volume in these cases.<ref>{{citation | last1 = Büeler | first1 = B. | last2 = Enge | first2 = A. | last3 = Fukuda | first3 = K. | doi = 10.1007/978-3-0348-8438-9_6 | chapter = Exact Volume Computation for Polytopes: A Practical Study | title = Polytopes — Combinatorics and Computation | pages = 131–154 | year = 2000 | isbn = 978-3-7643-6351-2 | citeseerx = 10.1.1.39.7700 }}</ref>