Editing Polyhedron (section)

===Surface area and lines inside polyhedra ===
The [[surface area]] of a polyhedron is the sum of the areas of its faces, for definitions of polyhedra for which the area of a face is well-defined.
The [[geodesic]] distance between any two points on the surface of a polyhedron measures the length of the shortest curve that connects the two points, remaining within the surface. By [[Alexandrov's uniqueness theorem]], every convex polyhedron is uniquely determined by the [[metric space]] of geodesic distances on its surface. However, non-convex polyhedra can have the same surface distances as each other, or the same as certain convex polyhedra.<ref>{{citation
 | last = Hartshorne | first = Robin | author-link = Robin Hartshorne
 | contribution = Example 44.2.3, the "punched-in icosahedron"
 | doi = 10.1007/978-0-387-22676-7
 | isbn = 0-387-98650-2
 | mr = 1761093
 | page = 442
 | publisher = Springer-Verlag, New York
 | series = Undergraduate Texts in Mathematics
 | title = Geometry: Euclid and beyond
 | year = 2000}}</ref>

When segment lines connect two vertices that are not in the same face, they form the [[diagonal|diagonal lines]].<ref name=pb>{{citation
 | last1 = Posamentier | first1 = Alfred S. 
 | last2 = Bannister | first2 = Robert L.
 | year = 2014
 | edition = 2nd
 | title = Geometry, Its Elements and Structure: Second Edition
 | url = https://books.google.com/books?id=XktMBAAAQBAJ&pg=PA543
 | page = 543
 | publisher = Dover Publications
 | isbn = 978-0-486-49267-4
}}</ref> However, not all polyhedra have diagonal lines, as in the family of [[Pyramid (geometry)|pyramids]],{{cn|date=February 2025}} [[Schönhardt polyhedron]] in which three diagonal lines lies entirely outside of it, and [[Császár polyhedron]] has no diagonal lines (rather, every pair of vertices is connected by an edge).<ref name=bagemihl>{{citation
 | last = Bagemihl | first = F. | authorlink = Frederick Bagemihl
 | journal = [[American Mathematical Monthly]]
 | pages = 411–413
 | title = On indecomposable polyhedra
 | volume = 55
 | year = 1948
 | doi = 10.2307/2306130
 | issue = 7
 | jstor = 2306130}}</ref>