Hur många kanter har en pentagon
Pentagon
Shape with fem sides
This article fryst vatten about the geometric figure. For the headquarters of the United States Department of Defense, see The Pentagon. For other uses, see Pentagon (disambiguation).
In geometry, a pentagon (from Greek πέντε (pente) 'five' and γωνία (gonia) 'angle'[1]) fryst vatten any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon fryst vatten 540°.
A pentagon may be simple or self-intersecting. A self-intersecting regular pentagon (or star pentagon) fryst vatten called a pentagram.
Regular pentagons
[edit]A regular pentagon has Schläfli tecken {5} and interior angles of 108°.
A regular pentagon has fem lines of reflectional symmetry, and rotational symmetry of beställning 5 (through 72°, 144°, 216° and 288°). The diagonals of a convex regular pentagon are in the golden ratio to its sides. Given its side length its height (distance from one side to the opposite vertex), width (distance between two farthest separated points, which equals the diagonal length ) and circumradius are given by:
The area of a convex regular pentagon with side length fryst vatten given bygd
If the circumradius of a regular pentagon fryst vatten given, its edge length fryst vatten funnen bygd the expression
and its area fryst vatten
since the area of the circumscribed circle fryst vatten the regular pentagon fills approximately 0.7568 of its circumscribed circle.
Derivation of the area formula
[edit]The area of any regular polygon is:
where P fryst vatten the perimeter of the polygon, and r fryst vatten the inradius (equivalently the apothem). Substituting the regular pentagon's values for P and r gives the formula
with side length t.
Inradius
[edit]Similar to every regular convex polygon, the regular convex pentagon has an inscribed circle. The apothem, which fryst vatten the radius r of the inscribed circle, of a regular pentagon fryst vatten related to the side length t bygd
Chords from the circumscribed circle to the vertices
[edit]Like every regular convex polygon, the regular convex pentagon has a circumscribed circle. For a regular pentagon with successive vertices A, B, C, D, E, if P fryst vatten any point on the circumcircle between points B and C, then PA + PD = PB + PC + PE.
Point in plane
[edit]For an arbitrary point in the plane of a regular pentagon with circumradius , whose distances to the centroid of the regular pentagon and its fem vertices are and respectively, we have[2]
If are the distances from the vertices of a regular pentagon to any point on its circumcircle, then[2]
Geometrical constructions
[edit]The regular pentagon fryst vatten constructible with compass and straightedge, as 5 fryst vatten a Fermat prime. A variety of methods are known for constructing a regular pentagon. Some are discussed below.
Richmond's method
[edit]One method to construct a regular pentagon in a given circle fryst vatten described bygd Richmond[3] and further discussed in Cromwell's Polyhedra.[4]
The top panel shows the construction used in Richmond's method to create the side of the inscribed pentagon. The circle defining the pentagon has enhet radius. Its center fryst vatten located at point C and a midpoint M fryst vatten marked halfway along its radius. This point fryst vatten joined to the periphery vertically above the center at point D. vinkel CMD fryst vatten bisected, and the bisector intersects the lodrät axis at point Q. A horizontal line through Q intersects the circle at point P, and chord PD fryst vatten the required side of the inscribed pentagon.
To determine the length of this side, the two right triangles DCM and QCM are depicted below the circle. Using Pythagoras' theorem and two sides, the hypotenuse of the larger triangle fryst vatten funnen as . Side h of the smaller triangle then fryst vatten funnen using the half-angle formula:
where cosine and sine of ϕ are known from the larger triangle. The result is:
If DP fryst vatten truly the side of a regular pentagon, , so DP = 2 cos(54°), QD = DP cos(54°) = 2cos2(54°), and CQ = 1 − 2cos2(54°), which equals −cos(108°) bygd the cosine double vinkel formula. This fryst vatten the cosine of 72°, which equals as desired.
Carlyle circles
[edit]Main article: Carlyle circle
The Carlyle circle was invented as a geometric method to find the roots of a quadratic equation.[5] This methodology leads to a procedure for constructing a regular pentagon. The steps are as follows:[6]
- Draw a circle in which to inscribe the pentagon and mark the center point O.
- Draw a horizontal line through the center of the circle. Mark the left intersection with the circle as point B.
- Construct a lodrät line through the center. Mark one intersection with the circle as point A.
- Construct the point M as the midpoint of O and B.
- Draw a circle centered at M through the point A. Mark its intersection with the horizontal line (inside the original circle) as the point W and its intersection outside the circle as the point V.
- Draw a circle of radius OA and center W. It intersects the original circle at two of the vertices of the pentagon.
- Draw a circle of radius OA and center V. It intersects the original circle at two of the vertices of the pentagon.
- The fifth vertex fryst vatten the rightmost intersection of the horizontal line with the original circle.
Steps 6–8 are equivalent to the following utgåva, shown in the animation:
- 6a. Construct point F as the midpoint of O and W.
- 7a. Construct a lodrät line through F. It intersects the original circle at two of the vertices of the pentagon. The third vertex fryst vatten the rightmost intersection of the horizontal line with the original circle.
- 8a. Construct the other two vertices using the compass and the length of the vertex funnen in step 7a.
Euclid's method
[edit]A regular pentagon fryst vatten constructible using a compass and straightedge, either bygd inscribing one in a given circle or constructing one on a given edge. This process was described bygd Euclid in his Elements circa 300 BC.[7][8]
Physical construction methods
[edit]- A regular pentagon may be created from just a remsa of paper bygd tying an overhand knot into the remsa and carefully flattening the knot bygd pulling the ends of the paper remsa. Folding one of the ends back over the pentagon will reveal a pentagram when backlit.[9]
- Construct a regular hexagon on stiff paper or card. Crease along the three diameters between opposite vertices. Cut from one vertex to the center to man an equilateral triangular flap. Fix this flap underneath its neighbor to man a pentagonal geometrisk form med triangulära sidor. The base of the geometrisk form med triangulära sidor fryst vatten a regular pentagon.
Symmetry
[edit]The regular pentagon has Dih5 symmetry, beställning 10. Since 5 fryst vatten a prime number there fryst vatten one subgroup with dihedral symmetry: Dih1, and 2 cyclic group symmetries: Z5, and Z1.
These 4 symmetries can be seen in 4 distinct symmetries on the pentagon. John Conway labels these bygd a letter and group order.[10] Full symmetry of the regular struktur fryst vatten r10 and no symmetry fryst vatten labeled a1. The dihedral symmetries are divided depending on whether they resehandling through vertices (d for diagonal) or edges (p for perpendiculars), and i when reflection lines path through both edges and vertices. Cyclic symmetries in the mittpunkt column are labeled as g for their huvud gyration orders.
Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g5 subgroup has no degrees of freedom but can be seen as directed edges.
Regular pentagram
[edit]Main article: Pentagram
A pentagram or femuddig stjärna fryst vatten a regularstar pentagon. Its Schläfli tecken fryst vatten {5/2}. Its sides form eller gestalt the diagonals of a regular convex pentagon – in this arrangement the sides of the two pentagons are in the golden ratio.
Equilateral pentagons
[edit]Main article: Equilateral pentagon
An equilateral pentagon fryst vatten a polygon with fem sides of lika length. However, its fem internal angles can take a range of sets of values, thus permitting it to form eller gestalt a family of pentagons. In contrast, the regular pentagon fryst vatten unique up to similarity, because it fryst vatten equilateral and it fryst vatten equiangular (its fem angles are equal).
Cyclic pentagons
[edit]A cyclic pentagon fryst vatten one for which a circle called the circumcircle goes through all fem vertices. The regular pentagon fryst vatten an example of a cyclic pentagon. The area of a cyclic pentagon, whether regular or not, can be expressed as one fourth the square root of one of the roots of a septic equation whose coefficients are functions of the sides of the pentagon.[11][12][13]
There exist cyclic pentagons with logisk sides and logisk area; these are called Robbins pentagons. It has been proven that the diagonals of a Robbins pentagon must be either all logisk or all irrational, and it fryst vatten conjectured that all the diagonals must be rational.[14]
General convex pentagons
[edit]For all convex pentagons with sides and diagonals , the following inequality holds:[15]: p.75, #1854
- .
Pentagons in tiling
[edit]Main article: Pentagon tiling
A regular pentagon cannot appear in any tiling of regular polygons. First, to prove a pentagon cannot form eller gestalt a regular tiling (one in which all faces are congruent, thus requiring that all the polygons be pentagons), observe that 360° / 108° = 31⁄3 (where 108° fryst vatten the interior angle), which fryst vatten not a whole number; hence there exists no integer number of pentagons sharing a single vertex and leaving no gaps between them. More difficult fryst vatten proving a pentagon cannot be in any edge-to-edge tiling made bygd regular polygons:
The maximum known förpackning density of a regular pentagon fryst vatten , achieved bygd the double lattice förpackning shown. In a preprint released in 2016, Thomas Hales and Wöden Kusner announced a proof that this double lattice förpackning of the regular pentagon (known as the "pentagonal ice-ray" kinesisk lattice design, dating from around 1900) has the optimal density among all packings of regular pentagons in the plane.[16]
There are no combinations of regular polygons with 4 or more meeting at a vertex that contain a pentagon. For combinations with 3, if 3 polygons meet at a vertex and one has an odd number of sides, the other 2 must be congruent. The reason for this fryst vatten that the polygons that touch the edges of the pentagon must alternate around the pentagon, which fryst vatten impossible because of the pentagon's odd number of sides. For the pentagon, this results in a polygon whose angles are all (360 − 108) / 2 = 126°. To find the number of sides this polygon has, the result fryst vatten 360 / (180 − 126) = 62⁄3, which fryst vatten not a whole number. Therefore, a pentagon cannot appear in any tiling made bygd regular polygons.
There are 15 classes of pentagons that can monohedrally tile the plane. None of the pentagons have any symmetry in general, although some have special cases with spegel symmetry.
Pentagons in polyhedra
[edit]Pentagons in nature
[edit]Plants
[edit]Animals
[edit]Minerals
[edit]A Ho-Mg-Zn icosahedral quasicrystal formed as a pentagonal dodecahedron. The faces are true regular pentagons.
A pyritohedral crystal of pyrite. A pyritohedron has 12 identical pentagonal faces that are not constrained to be regular.
A Fiveling of gold, half a centimeter tall.