page_title: Equilateral pentagon
section_title: Internal angles
hierarchical_section_title: Equilateral pentagon / Internal angles
context_page_description: In geometry an equilateral pentagon is a polygon with five sides of equal length. Its five internal angles, in turn, can take a range of sets of values, thus permitting it to form a family of pentagons. The requirement is that all angles must add up to 540 degrees and must be between 0 and 360 degrees but not equal to 180 degrees. In contrast, the regular pentagon is unique, because it is equilateral and moreover it is equiangular.
  Four intersecting equal circles arranged in a closed chain are sufficient to determine a convex equilateral pentagon. Each circle's center is one of four vertices of the pentagon. The remaining vertex is determined by one of the intersection points of the first and the last circle of the chain.
  It is possible to describe the five angles of any convex equilateral pentagon with only two angles α and β, provided α ≥ β and δ is the smallest of the other angles. Thus the general equilateral pentagon can be regarded as a bivariate function f where the rest of the angles can be obtained by using trigonometric relations. The equilateral pentagon described in this manner will be unique up to a rotation in the plane.
context_section_description: When the equilateral pentagon is dissected into triangles, two of them appear as isosceles (triangles in orange and blue) while the other one is more general (triangle in green). We assume that we are given the adjacent angles  and . 
  According to the law of sines the length of the line dividing the green and blue triangles is:
  The square of the length of the line dividing the orange and green triangles is:
  According to the law of cosines, the cosine of δ can be seen from the figure:
  Simplifying, δ is obtained as function of α and β:
  The remaining angles of the pentagon can be found geometrically: The remaining angles of the orange and blue triangles are readily found by noting that two angles of an isosceles triangle are equal while all three angles sum to 180°. Then  and the two remaining angles of the green triangle can be found from four equations stating that the sum of the angles of the pentagon is 540°, the sum of the angles of the green triangle is 180°, the angle  is the sum of its three components, and the angle  is the sum of its two components.
  A cyclic pentagon is equiangular if and only if it has equal sides and thus is regular. Likewise, a tangential pentagon is equilateral if and only if it has equal angles and thus is regular.
caption: Equilateral Pentagon dissected into 3 triangles which helps to calculate the value of angle δ as a function of α and β.