Calculations at a regular pentagon, a polygon with 5 vertices, it has 5 equal sides and angles.
Enter one value and choose the number of decimal places. Then click Calculate.
Formulas:
d = a / 2 * ( 1 + √5 )
h = a / 2 * √ 5 + 2 * √5
p = 5 * a
A = a² / 4 * √ 25 + 10 * √5
rc = a / 10 * √ 50 + 10 * √5
ri = a / 10 * √ 25 + 10 * √5
Angle: 108°
5 diagonals
Edge length, diagonals, height, perimeter and radius have the same unit (e.g. meter), the area has this unit squared (e.g. square meter).
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Heights, bisecting lines and median lines coincide, these intersect at the centroid, which is also circumcircle and incircle center. To this point, the regular pentagon is rotationally symmetric at a rotation of 72° or multiples of this. Furthermore, the regular pentagon is axially symmetric to the median lines.
In a regular pentagon, the golden ratio is found in the ratio of the lengths of a diagonal to a side. All five diagonals together form the pentagram. The regular pentagon forms the side surfaces of one of the five Platonic solids, the dodecahedron. Furthermore, this shape can be found in four of the thirteen Archimedean solids, the icosidodecahedron, the truncated icosahedron, as well as in the rhombenicosidodecahedron and the snub dodecahedrons.
The pentagon theorem states that a spatial pentagon with equal sides and angles always lies in one plane. So there is no such structure that spans a three-dimensional space. This does not apply to all other regular polygons, except of course for the triangle, which in its general form is always on one plane anyway. The pentagon theorem was proven in the second half of the 20th century.
The regular Pentagon appears frequently in architecture. The most famous example is of course the headquarters building of the US Department of Defense near Washington. Modern fortifications were often designed in a pentagonal shape.
