It has six sides, six edges and six vertices.Concave Hexagonįor concave hexagons, at least one of its interior angles must be greater than 180° and at least one of its vertex points inwards. Convex hexagons can be regular or irregular. Convex Hexagonįor convex hexagons, all of its interior angles must be less than 180 degrees, and all the vertices are pointed outwards. If the successive sides of a cyclic hexagon are a, b, c, d, e, f, then the three main diagonals intersect in a single point if and only if ace = bdf.Classification of hexagons based on their angles 1.

The Lemoine hexagon is a cyclic hexagon (one inscribed in a circle) with vertices given by the six intersections of the edges of a triangle and the three lines that are parallel to the edges that pass through its symmedian point. Pascal's theorem (also known as the 'Hexagrammum Mysticum Theorem') states that if an arbitrary hexagon is inscribed in any conic section, and pairs of opposite sides are extended until they meet, the three intersection points will lie on a straight line, the 'Pascal line' of that configuration. In addition to the regular hexagon, which determines a unique tessellation of the plane, any irregular hexagon which satisfies the Conway criterion will tile the plane. Self-intersecting hexagons with regular vertices There are six self-crossing hexagons with the vertex arrangement of the regular hexagon: This pattern repeats within the rhombitrihexagonal tiling.Ī self-intersecting hexagon ( star polygon) This pattern repeats within the regular triangular tiling.Ī regular hexagon can be extended into a regular dodecagon by adding alternating squares and equilateral triangles around it.

A regular hexagon can be dissected into six equilateral triangles by adding a center point. A regular hexagon can be stellated with equilateral triangles on its edges, creating a hexagram. The common length of the sides equals the radius of the circumscribed circle or circumcircle, which equals 2 3. It is bicentric, meaning that it is both cyclic (has a circumscribed circle) and tangential (has an inscribed circle).

Transfer the line segment AB four times on the circumscribed circle and connect the corner points.Ī regular hexagon is defined as a hexagon that is both equilateral and equiangular. When the side length AB is given, drawing a circular arc from point A and point B gives the intersection M, the center of the circumscribed circle.

Perfect hexagon measurements