|Symmetry group||D5d, [2+,10], (2*5), order 20|
|Rotation group||D5, [2,5]+, (225), order 10|
|Dual polyhedron||pentagonal antiprism|
The pentagonal trapezohedron or deltohedron is the third in an infinite series of face-transitive polyhedra which are dual polyhedra to the antiprisms. It has ten faces (i.e., it is a decahedron) which are congruent kites.
The pentagonal trapezohedron was patented for use as a gaming die in 1906. It is convenient for role-playing games that use percentile-based skills; however, it is not strictly necessary since the outcome of a twenty-sided die can be divided ten ways and is sometimes preferred due to its regular shape.
Subsequent patents on ten-sided dice have made minor refinements to the basic design by rounding or truncating the edges. This enables the die to tumble so that the outcome is less predictable. One such refinement became notorious at the 1980 Gen Con when the patent was incorrectly thought to cover ten-sided dice in general.
Ten sided dice are commonly numbered from 0 to 9, as this allows two to be rolled in order to easily obtain a percentile result. Where one die represents the 'tens', the other represents 'units' therefore a result of 7 on the former and 0 on the latter would be combined to produce 70. A result of double-zero is commonly interpreted as 100. Some ten-sided dice (often called 'Percentile Dice') are sold in sets of two where one is numbered from 0 to 9 and the other from 0 to 90 in increments of 10, thus making it impossible to misinterpret which one is the tens and which the units die. Ten sided dice may also be numbered 1 to 10 for use in games where a random number in this range is desirable, or the zero may be interpreted as 10 in this situation.
A fairly consistent arrangement of the faces on ten-digit dice has been observed. If one holds such a die between one's fingers at two of the vertices such that the even numbers are on top, and reads the numbers from left to right in a zigzag pattern, the sequence obtained is 0, 7, 4, 1, 6, 9, 2, 5, 8, 3, and back to 0. The even and odd digits are divided among the two opposing "caps" of the die, and each pair of opposite faces adds to nine.
- U.S. Patent 809,293
- Greg Peterson about Gen Con 1980: The big news of the year was that someone had 'invented' the ten-sided die.
- Cundy H.M and Rollett, A.P. Mathematical models, 2nd Edn. Oxford University Press (1961), (3rd edition 1989) p. 117
-  Generalized formula of uniform polyhedron (trapezohedron) having 2n congruent right kite faces
- Weisstein, Eric W., "Trapezohedron", MathWorld.
- Virtual Reality Polyhedra www.georgehart.com: The Encyclopedia of Polyhedra