Editing Cube (section)

=== Measurement and other metric properties ===
[[File:Cube diagonals.svg|thumb|upright=0.6|A face diagonal in red and space diagonal in blue]]
Given a cube with edge length <math> a </math>. The [[face diagonal]] of a cube is the [[diagonal]] of a square <math> a\sqrt{2} </math>, and the [[space diagonal]] of a cube is a line connecting two vertices that is not in the same face, formulated as <math> a \sqrt{3} </math>. Both formulas can be determined by using [[Pythagorean theorem]]. The surface area of a cube <math> A </math> is six times the area of a square:{{r|khattar}}
<math display="block"> A = 6a^2. </math>
The volume of a cuboid is the product of its length, width, and height. Because all the edges of a cube are equal in length, the formula for the volume of a cube as the third power of its side length, leading to the use of the term ''[[Cube (algebra)|cubic]]'' to mean raising any number to the third power:{{r|thomson|khattar}}
<math display="block"> V = a^3. </math>

[[File:Prince Ruperts cube.png|thumb|upright=0.6|[[Prince Rupert's cube]]]]
One special case is the [[unit cube]], so named for measuring a single [[unit of length]] along each edge. It follows that each face is a [[unit square]] and that the entire figure has a volume of 1 cubic unit.{{r|ball|hr-w}} [[Prince Rupert's cube]], named after [[Prince Rupert of the Rhine]], is the largest cube that can pass through a hole cut into the unit cube, despite having sides approximately 6% longer.{{r|sriraman}} A polyhedron that can pass through a copy of itself of the same size or smaller is said to have the [[Rupert property]].{{r|jwy}} A geometric problem of [[doubling the cube]]&mdash;alternatively known as the ''Delian problem''&mdash;requires the construction of a cube with a volume twice the original by using a [[compass and straightedge]] solely. Ancient mathematicians could not solve this old problem until the French mathematician [[Pierre Wantzel]] in 1837 proved it was impossible.{{r|lutzen}}

The cube has three types of [[closed geodesic]]s. The closed geodesics are paths on a cube's surface that are locally straight. In other words, they avoid the vertices of the polyhedron, follow line segments across the faces that they cross, and form [[complementary angle]]s on the two incident faces of each edge that they cross. Two of its types are planar. The first type lies in a plane parallel to any face of the cube, forming a square, with the length being equal to the perimeter of a face, four times the length of each edge. The second type lies in a plane perpendicular to the long diagonal, forming a regular hexagon; its length is <math> 3 \sqrt 2 </math> times that of an edge. The third type is a non-planar hexagon<!--, with the length being <math> 20 </math> (How is this derived? What is the unit of length? -->.{{r|fuchs}}