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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.
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, moreover, it is equiangular (its five angles are equal).
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.
This process was described by Euclid in his Elements circa 300 BC.
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.
In a Robbins pentagon, either all diagonals are rational or all are irrational, and it is conjectured that all the diagonals must be rational.
complete graph is often drawn as a regular pentagon with all 10 edges connected.