Pentagon | |
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Edges and vertices | 5 |
Internal angle (degrees) | 108° (if equiangular, including regular) |
In geometry, a pentagon (from the Greek πέντε pente and γωνία gonia, meaning five and angle^{[1]}) is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.
A pentagon may be simple or self-intersecting. A self-intersecting regular pentagon (or star pentagon) is called a pentagram.
Regular pentagon | |
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Type | Regular polygon |
Edges and vertices | 5 |
Schläfli symbol | {5} |
Coxeter diagram | |
Symmetry group | Dihedral (D_{5}), order 2×5 |
Internal angle (degrees) | 108° |
Dual polygon | Self |
Properties | Convex, cyclic, equilateral, isogonal, isotoxal |
A regular pentagon has Schläfli symbol {5} and interior angles of 108°.
A regular pentagon has five lines of reflectional symmetry, and rotational symmetry of order 5 (through 72°, 144°, 216° and 288°). The diagonals of a convex regular pentagon are in the golden ratio to its sides. Its height (distance from one side to the opposite vertex) and width (distance between two farthest separated points, which equals the diagonal length) are given by
where R is the radius of the circumcircle.
The area of a convex regular pentagon with side length t is given by
A pentagram or pentangle is a regular star pentagon. Its Schläfli symbol is {5/2}. Its sides form the diagonals of a regular convex pentagon – in this arrangement the sides of the two pentagons are in the golden ratio.
When a regular pentagon is circumscribed by a circle with radius R, its edge length t is given by the expression
and its area is
since the area of the circumscribed circle is the regular pentagon fills approximately 0.7568 of its circumscribed circle.
The area of any regular polygon is:
where P is the perimeter of the polygon, and r is the inradius (equivalently the apothem). Substituting the regular pentagon's values for P and r gives the formula
with side length t.
Like every regular convex polygon, the regular convex pentagon has an inscribed circle. The apothem, which is the radius r of the inscribed circle, of a regular pentagon is related to the side length t by
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 is any point on the circumcircle between points B and C, then PA + PD = PB + PC + PE.
For an arbitrary point in the plane of a regular pentagon with circumradius , whose distances to the centroid of the regular pentagon and its five 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]}
The regular pentagon is constructible with compass and straightedge, as 5 is a Fermat prime. A variety of methods are known for constructing a regular pentagon. Some are discussed below.
One method to construct a regular pentagon in a given circle is described by 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 unit radius. Its center is located at point C and a midpoint M is marked halfway along its radius. This point is joined to the periphery vertically above the center at point D. Angle CMD is bisected, and the bisector intersects the vertical axis at point Q. A horizontal line through Q intersects the circle at point P, and chord PD is 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 is found as . Side h of the smaller triangle then is found using the half-angle formula:
where cosine and sine of ϕ are known from the larger triangle. The result is:
With this side known, attention turns to the lower diagram to find the side s of the regular pentagon. First, side a of the right-hand triangle is found using Pythagoras' theorem again:
Then s is found using Pythagoras' theorem and the left-hand triangle as:
The side s is therefore:
a well-established result.^{[5]} Consequently, this construction of the pentagon is valid.
Main article: Carlyle circle |
The Carlyle circle was invented as a geometric method to find the roots of a quadratic equation.^{[6]} This methodology leads to a procedure for constructing a regular pentagon. The steps are as follows:^{[7]}
Steps 6–8 are equivalent to the following version, shown in the animation:
This follows quickly from the knowledge that twice the sine of 18 degrees is the reciprocal golden ratio, which we know geometrically from the triangle with angles of 72,72,36 degrees. From trigonometry, we know that the cosine of twice 18 degrees is 1 minus twice the square of the sine of 18 degrees, and this reduces to the desired result with simple quadratic arithmetic.
The regular pentagon according to the golden ratio, dividing a line segment by exterior division
A regular pentagon is constructible using a compass and straightedge, either by inscribing one in a given circle or constructing one on a given edge. This process was described by Euclid in his Elements circa 300 BC.^{[8]}^{[9]}
A direct method using degrees follows:
After forming a regular convex pentagon, if one joins the non-adjacent corners (drawing the diagonals of the pentagon), one obtains a pentagram, with a smaller regular pentagon in the center. Or if one extends the sides until the non-adjacent sides meet, one obtains a larger pentagram. The accuracy of this method depends on the accuracy of the protractor used to measure the angles.
The regular pentagon has Dih_{5} symmetry, order 10. Since 5 is a prime number there is one subgroup with dihedral symmetry: Dih_{1}, and 2 cyclic group symmetries: Z_{5}, and Z_{1}.
These 4 symmetries can be seen in 4 distinct symmetries on the pentagon. John Conway labels these by a letter and group order.^{[10]} Full symmetry of the regular form is r10 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass 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 middle column are labeled as g for their central 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.
Main article: Equilateral pentagon |
An equilateral pentagon is a polygon with five sides of equal length. However, its five internal angles can take a range of sets of values, thus permitting it to form a family of pentagons. In contrast, the regular pentagon is unique up to similarity, because it is equilateral and it is equiangular (its five angles are equal).
A cyclic pentagon is one for which a circle called the circumcircle goes through all five vertices. The regular pentagon is 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 rational sides and rational area; these are called Robbins pentagons. In a Robbins pentagon, either all diagonals are rational or all are irrational, and it is conjectured that all the diagonals must be rational.^{[14]}
For all convex pentagons, the sum of the squares of the diagonals is less than 3 times the sum of the squares of the sides.^{[15]}^{:p.75,#1854}
The K_{5} complete graph is often drawn as a regular pentagon with all 10 edges connected. This graph also represents an orthographic projection of the 5 vertices and 10 edges of the 5-cell. The rectified 5-cell, with vertices at the mid-edges of the 5-cell is projected inside a pentagon.
5-cell (4D) |
Rectified 5-cell (4D) |
Main article: Pentagon tiling |
A regular pentagon cannot appear in any tiling of regular polygons. First, to prove a pentagon cannot form 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° Is the interior angle), which is not a whole number; hence there exists no integer number of pentagons sharing a single vertex and leaving no gaps between them. More difficult is proving a pentagon cannot be in any edge-to-edge tiling made by regular polygons:
The maximum known packing density of a regular pentagon is approximately 0.921, achieved by the double lattice packing shown. In a preprint released in 2016, Thomas Hales and Wöden Kusner announced a proof that the double lattice packing of the regular pentagon (which they call the "pentagonal ice-ray" packing, and which they trace to the work of Chinese artisans in 1900) has the optimal density among all packings of regular pentagons in the plane.^{[16]} As of 2020^{[update]}, their proof has not yet been refereed and published.
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 is that the polygons that touch the edges of the pentagon must alternate around the pentagon, which is 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 is 360 / (180 − 126) = 62⁄3, which is not a whole number. Therefore, a pentagon cannot appear in any tiling made by 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 mirror symmetry.
1 | 2 | 3 | 4 | 5 |
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6 | 7 | 8 | 9 | 10 |
11 | 12 | 13 | 14 | 15 |
I_{h} | T_{h} | T_{d} | O | I | D_{5d} |
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Dodecahedron | Pyritohedron | Tetartoid | Pentagonal icositetrahedron | Pentagonal hexecontahedron | Truncated trapezohedron |