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Regular pentagon  

A regular pentagon  
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 
Regular pentagon  

A regular pentagon  
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 
In geometry, a pentagon (from pente and gonia, which is Greek for five and angle) is any fivesided polygon. A pentagon may be simple or selfintersecting. The sum of the internal angles in a simple pentagon is 540°. A pentagram is an example of a selfintersecting pentagon.
In a regular pentagon, all sides are equal in length and each interior angle is 108°. A regular pentagon has five lines of reflectional symmetry, and rotational symmetry of order 5 (through 72°, 144°, 216° and 288°). Its Schläfli symbol is {5}. The diagonals of a regular pentagon are in golden ratio to its sides.
The area of a regular convex 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 inscribed in a circle with radius R, its edge length t is given by the expression
The area of any regular polygon is:
where P is the perimeter of the polygon, a is the apothem. One can then substitute the respective values for P and a, which makes the formula:
with t as the given side length. Then we can then rearrange the formula as:
and then, we combine the two terms to get the final formula, which is:
The diagonals of a regular pentagon (hereby represented by D) can be calculated based upon the golden ratio φ and the known side T (see discussion of the pentagon in Golden ratio):
Accordingly:
If a regular pentagon with successive vertices A, B, C, D, E is inscribed in a circle, and if P is any point on that circle between points B and C, then PA + PD = PB + PC + PE.
A variety of methods are known for constructing a regular pentagon. Some are discussed below.
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.^{[1]}
One method to construct a regular pentagon in a given circle is described by Richmond^{[2]} and further discussed in Cromwell's "Polyhedra."^{[3]}
The top panel describes the construction used in the animation above 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 halfangle 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 righthand triangle is found using Pythagoras' theorem again:
Then s is found using Pythagoras' theorem and the lefthand triangle as:
The side s is therefore:
a well established result.^{[4]} Consequently, this construction of the pentagon is valid.
An alternative method is this:
See 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]}
Steps 68 are equivalent to the following version, shown in the animation:
6a. Construct point F as the midpoint of O and W.
7a. Construct a vertical line through F. It intersects the original circle at two of the vertices of the pentagon. The third vertex is 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 found in step 7a.
A direct method using degrees follows:
After forming a regular convex pentagon, if one joins the nonadjacent 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 nonadjacent sides meet, one obtains a larger pentagram. The accuracy of this method depends on the accuracy of the protractor used to measure the angles.
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.^{[7]}^{[8]}^{[9]}
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.^{[10]}
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 5cell. The rectified 5cell, with vertices at the midedges of the 5cell is projected inside a pentagon.
5cell (4D)  Rectified 5cell (4D) 
Pentagonal crosssection of okra.
Morning glories, like many other flowers, have a pentagonal shape.
The gynoecium of an apple contains five carpels, arranged in a fivepointed star
Starfruit is another fruit with fivefold symmetry.
A sea star. Many echinoderms have fivefold radial symmetry.
An illustration of brittle stars, also echinoderms with a pentagonal shape.
The Pentagon, headquarters of the United States Department of Defense.
Home plate of a baseball field
A pentagon cannot appear in any tiling made by regular polygons. To prove a pentagon cannot form a regular tiling (one in which all faces are congruent), observe that 360 / 108 = 3^{1}⁄_{3}, which is not a whole number. More difficult is proving a pentagon cannot be in any edgetoedge tiling made by regular polygons:
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) = 6^{2}⁄_{3}, which is not a whole number. Therefore, a pentagon cannot appear in any tiling made by regular polygons.
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