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Concept
Pentagon
Summary
In geometry, a pentagon (from the Greek πέντε pente meaning five and γωνία gonia meaning angle) is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.
A pentagon may be simple or self-intersecting. A self-intersecting regular pentagon (or star pentagon) is called a pentagram.
A regular pentagon has Schläfli symbol {5} and interior angles of 108°.
A regular pentagon has five lines of reflectional symmetry, and rotational symmetry of order 5 (through 72°, 144°, 216° and 288°). The diagonals of a convex regular pentagon are in the golden ratio to its sides. Given its side length its height (distance from one side to the opposite vertex), width (distance between two farthest separated points, which equals the diagonal length ) and circumradius are given by:
The area of a convex regular pentagon with side length is given by
If the circumradius of a regular pentagon is given, its edge length is found by the expression
and its area is
since the area of the circumscribed circle is the regular pentagon fills approximately 0.7568 of its circumscribed circle.
The area of any regular polygon is:
where P is the perimeter of the polygon, and r is the inradius (equivalently the apothem). Substituting the regular pentagon's values for P and r gives the formula
with side length t.
Similar to 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. For a regular pentagon with successive vertices A, B, C, D, E, if P is any point on the circumcircle between points B and C, then PA + PD = PB + PC + PE.
For an arbitrary point in the plane of a regular pentagon with circumradius , whose distances to the centroid of the regular pentagon and its five vertices are and
respectively, we have
If are the distances from the vertices of a regular pentagon to any point on its circumcircle, then
The regular pentagon is constructible with compass and straightedge, as 5 is a Fermat prime.
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