![]() The width w is the distance between two opposite vertices of the regular pentagon (the length of its diagonal). Substituting the value of \theta to the last expressions we get the following approximations: This is a right triangle since by definition the incircle is tangential to all sides of the polygon. To this end, we will examine the triangle with sides the circumradius, the inradius and half the pentagon edge, as highlighted in the figure below. We will try to find the relationships between the side length a of the regular pentagon and its circumradius R_c and inradius R_i. Circumcircle and incircle of a regular pentagon The following figure depicts both circumscribed circle of the regular pentagon and the inscribed one. The radius of incircle, R_i, is usually called inradius. Its center is the same with the center of the circumcircle and it is tangent to all five sides of the regular pentagon. This is the so called inscribed circle or incircle. The radius of circumcircle, R_c, is usually called circumradius.Īnother circle can also be drawn, that touches tangentially all five edges of the regular pentagon at the midpoints (also a common characteristic of all regular polygons). This is the so called cirmuscribed circle or circumcircle of the regular pentagon (indeed this is a common characteristic of all regular polygons). The center of this circle is also the center of the pentagon, where all the symmetry axes are intersecting also. It is possible to draw a circle that passes through all the five vertices of the regular pentagon. The regular pentagon is divided into five identical isosceles triangles having a common vertex, the polygon center. Interior and central angle of a regular pentagon In other words \varphi and \theta are supplementary. ![]() It is not coincidence that the sum of interior and central angles is 180°: That is also the half of the interior angle \varphi, (108°/2=54°). The central angle, \theta, of each triangle is:įocusing on one of the five triangles, its two remaining angles are identical and equal to 54°, so that the sum of all angles in the triangle is 180°, (72°+54°+54°). Therefore, the interior angle, \varphi, of a regular pentagon should be 108°:įive identical, isosceles, triangles are defined if we draw straight lines from the center of the regular pentagon towards each one of its vertices. It is also a common property of all pentagons that the sum of their interior angles is always 540°, as explained previously. Axes of symmetry of regular pentagon Interior angle and central angleīy definition the interior angles of a regular pentagon are equal. This is in fact its center of gravity or centroid. All axes of symmetry intersect at a common point, the center of the regular pentagon. Each one of them passes through a vertex of the pentagon and the middle of the opposite edge, as shown in the following drawing. A pentagon can be divided into three triangles Properties of regular pentagons SymmetryĪ regular pentagon has five axes of symmetry. In a single triangle the sum of internal angles is 180°, therefore, for 3 triangles, positioned side by side, the internal angles should measure up to 3x180°=540°. There are many different ways to draw lines between the vertices, resulting in different triangles, however their count is always three. If we try to draw straight lines between all vertices, avoiding any intersections, we divide the pentagon into three individual triangles. It can be easily proved by decomposing the pentagon to individual, non overlapping triangles. This is true for either regular or irregular pentagons, convex or concave. ![]() The sum of the internal angles of a pentagon is constant and equal to 540°. Any pentagon that is not regular is called irregular. The next figure illustrates the classification of pentagons, also presenting equilateral ones that are concave. In that case the pentagon is called equilateral. Having only the sides equal is not adequate, because the pentagon can be concave with equal sides. A pentagon is regular when all its sides and interior angles are equal. ![]() A concave polygon, to the contrary, does have one or more of its interior angles larger than 180°. When convex, the pentagon (or any closed polygon in that matter) does have all its interior angles lower than 180°. A pentagon may be either convex or concave, as depicted in the next figure. Pentagon is a polygon with five sides and five vertices.
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