Equilateral pentagon

In geometry, an equilateral pentagon is a polygon in the Euclidean plane with five sides of equal length.

Its five vertex 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, because it is equilateral and moreover it is equiangular (its five angles are equal; the measure is 108 degrees).

Four intersecting equal circles arranged in a closed chain are sufficient to determine a convex equilateral pentagon.

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°.

Given that we rule out the pentagons that intersect themselves once, we can plot the rest as a function of two variables in the two-dimensional plane.

The equation δ = β as a curve divides the plane into different sections (north border shown in blue).

Both borders enclose a continuous region of the plane whose points map to unique equilateral pentagons.

Inside the region of unique mappings there are three types of pentagons: stellated, concave and convex, separated by new borders.

So, in the mapping, the line 2α + β = 180° (shown in orange at the north) is the border between the regions of stellated and non-stellated pentagons.