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1. Pentagon – In geometry, a pentagon is any five-sided polygon or 5-gon. The sum of the angles in a simple pentagon is 540°. A pentagon may be simple or self-intersecting, a self-intersecting regular pentagon is called a pentagram. A regular pentagon has Schläfli symbol and interior angles are 108°, a regular pentagon has five lines of reflectional symmetry, and rotational symmetry of order 5. The diagonals of a regular pentagon are in the golden ratio to its sides. The area of a regular convex pentagon with side length t is given by A = t 225 +1054 =5 t 2 tan ⁡4 ≈1.720 t 2. A pentagram or pentangle is a regular star pentagon and its sides form the diagonals of a regular convex pentagon – in this arrangement the sides of the two pentagons are in the golden ratio. The area of any polygon is, A =12 P r where P is the perimeter of the polygon. Substituting the regular pentagons values for P and r gives the formula A =12 ×5 t × t tan ⁡2 =5 t 2 tan ⁡4 with side length t, like every regular convex polygon, the regular convex pentagon has an inscribed circle. The apothem, which is the r of the inscribed circle. Like every regular polygon, the regular convex pentagon has a circumscribed circle. For a regular pentagon with successive vertices A, B, C, D, E, the regular pentagon is constructible with compass and straightedge, as 5 is a Fermat prime. A variety of methods are known for constructing a regular pentagon, one method to construct a regular pentagon in a given circle is described by Richmond and further discussed in Cromwells Polyhedra. The top panel shows the construction used in Richmonds method to create the side of the inscribed pentagon, the circle defining the pentagon has unit radius. Its center is located at point C and a midpoint M is marked halfway along its radius and this point is joined to the periphery vertically above the center at point D. Angle CMD is bisected, and the bisector intersects the axis at point Q. A horizontal line through Q intersects the circle at point P, to determine the length of this side, the two right triangles DCM and QCM are depicted below the circle. Using Pythagoras theorem and two sides, the hypotenuse of the triangle is found as 5 /2

2. Hexagon – In geometry, a hexagon is a six sided polygon or 6-gon. The total of the angles of any hexagon is 720°. A regular hexagon has Schläfli symbol and can also be constructed as an equilateral triangle, t. A regular hexagon is defined as a hexagon that is both equilateral and equiangular and it is bicentric, meaning that it is both cyclic and tangential. The common length of the sides equals the radius of the circumscribed circle, all internal angles are 120 degrees. A regular hexagon has 6 rotational symmetries and 6 reflection symmetries, the longest diagonals of a regular hexagon, connecting diametrically opposite vertices, are twice the length of one side. Like squares and equilateral triangles, regular hexagons fit together without any gaps to tile the plane, the cells of a beehive honeycomb are hexagonal for this reason and because the shape makes efficient use of space and building materials. The Voronoi diagram of a triangular lattice is the honeycomb tessellation of hexagons. It is not usually considered a triambus, although it is equilateral, the maximal diameter, D is twice the maximal radius or circumradius, R, which equals the side length, t. The minimal diameter or the diameter of the circle, d, is twice the minimal radius or inradius. If a regular hexagon has successive vertices A, B, C, D, E, F, the regular hexagon has Dih6 symmetry, order 12. There are 3 dihedral subgroups, Dih3, Dih2, and Dih1, and 4 cyclic subgroups, Z6, Z3, Z2 and these symmetries express 9 distinct symmetries of a regular hexagon. John Conway labels these by a letter and group order, r12 is full symmetry, and a1 is no symmetry. These two forms are duals of each other and have half the order of the regular hexagon. The i4 forms are regular hexagons flattened or stretched along one symmetry direction and it can be seen as an elongated rhombus, while d2 and p2 can be seen as horizontally and vertically elongated kites. G2 hexagons, with sides parallel are also called hexagonal parallelogons. Each subgroup symmetry allows one or more degrees of freedom for irregular forms, only the g6 subgroup has no degrees of freedom but can seen as directed edges. Hexagons of symmetry g2, i4, and r12, as parallelogons can tessellate the Euclidean plane by translation, other hexagon shapes can tile the plane with different orientations

3. Octagon – In geometry, an octagon is an eight-sided polygon or 8-gon. A regular octagon has Schläfli symbol and can also be constructed as a truncated square, t. A truncated octagon, t is a hexadecagon, t, the sum of all the internal angles of any octagon is 1080°. As with all polygons, the external angles total 360°, the midpoint octagon of a reference octagon has its eight vertices at the midpoints of the sides of the reference octagon. A regular octagon is a figure with sides of the same length. It has eight lines of symmetry and rotational symmetry of order 8. A regular octagon is represented by the Schläfli symbol, the internal angle at each vertex of a regular octagon is 135°. The area of an octagon of side length a is given by A =2 cot ⁡ π8 a 2 =2 a 2 ≃4.828 a 2. In terms of the circumradius R, the area is A =4 sin ⁡ π4 R2 =22 R2 ≃2.828 R2. In terms of the r, the area is A =8 tan ⁡ π8 r 2 =8 r 2 ≃3.314 r 2. These last two coefficients bracket the value of pi, the area of the unit circle. The area can also be expressed as A = S2 − a 2, where S is the span of the octagon, or the second-shortest diagonal, and a is the length of one of the sides, or bases. This is easily proven if one takes an octagon, draws a square around the outside and then takes the corner triangles and places them with right angles pointed inward, the edges of this square are each the length of the base. Given the length of a side a, the span S is S = a 2 + a + a 2 = a ≈2.414 a. The area is then as above, A =2 − a 2 =2 a 2 ≈4.828 a 2, expressed in terms of the span, the area is A =2 S2 ≈0.828 S2. Another simple formula for the area is A =2 a S, more often the span S is known, and the length of the sides, a, is to be determined, as when cutting a square piece of material into a regular octagon. From the above, a ≈ S /2.414, the two end lengths e on each side, as well as being e = a /2, may be calculated as e = /2. The circumradius of the octagon in terms of the side length a is R = a

4. Dodecagon – In geometry, a dodecagon or 12-gon is any twelve-sided polygon. A regular dodecagon is a figure with sides of the same length. It has twelve lines of symmetry and rotational symmetry of order 12. A regular dodecagon is represented by the Schläfli symbol and can be constructed as a hexagon, t, or a twice-truncated triangle. The internal angle at each vertex of a regular dodecagon is 150°, as 12 =22 ×3, regular dodecagon is constructible using compass and straightedge, Coxeter states that every parallel-sided 2m-gon can be divided into m/2 rhombs. For the dodecagon, m=6, and it can be divided into 15 rhombs and this decomposition is based on a Petrie polygon projection of a 6-cube, with 15 of 240 faces. One of the ways the mathematical manipulative pattern blocks are used is in creating a number of different dodecagons, the regular dodecagon has Dih12 symmetry, order 24. There are 15 distinct subgroup dihedral and cyclic symmetries, each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g12 subgroup has no degrees of freedom but can seen as directed edges, the interior of such an dodecagon is not generally defined. A skew zig-zag dodecagon has vertices alternating between two parallel planes, a regular skew dodecagon is vertex-transitive with equal edge lengths. In 3-dimensions it will be a zig-zag skew dodecagon and can be seen in the vertices and side edges of a antiprism with the same D5d, symmetry. The dodecagrammic antiprism, s and dodecagrammic crossed-antiprism, s also have regular skew dodecagons, the regular dodecagon is the Petrie polygon for many higher-dimensional polytopes, seen as orthogonal projections in Coxeter planes. Examples in 4 dimensionare the 24-cell, snub 24-cell, 6-6 duoprism, in 6 dimensions 6-cube, 6-orthoplex,221,122. It is also the Petrie polygon for the grand 120-cell and great stellated 120-cell, a dodecagram is a 12-sided star polygon, represented by symbol. There is one regular star polygon, using the same vertices, but connecting every fifth point. There are also three compounds, is reduced to 2 as two hexagons, and is reduced to 3 as three squares, is reduced to 4 as four triangles, and is reduced to 6 as six degenerate digons. Deeper truncations of the regular dodecagon and dodecagrams can produce intermediate star polygon forms with equal spaced vertices. A truncated hexagon is a dodecagon, t=, a quasitruncated hexagon, inverted as, is a dodecagram, t=

5. Rectangle – In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as a quadrilateral, since equiangular means that all of its angles are equal. It can also be defined as a parallelogram containing a right angle, a rectangle with four sides of equal length is a square. The term oblong is occasionally used to refer to a non-square rectangle, a rectangle with vertices ABCD would be denoted as ABCD. The word rectangle comes from the Latin rectangulus, which is a combination of rectus and angulus, a crossed rectangle is a crossed quadrilateral which consists of two opposite sides of a rectangle along with the two diagonals. It is a case of an antiparallelogram, and its angles are not right angles. Other geometries, such as spherical, elliptic, and hyperbolic, have so-called rectangles with sides equal in length. Rectangles are involved in many tiling problems, such as tiling the plane by rectangles or tiling a rectangle by polygons, a convex quadrilateral with successive sides a, b, c, d whose area is 12. A rectangle is a case of a parallelogram in which each pair of adjacent sides is perpendicular. A parallelogram is a case of a trapezium in which both pairs of opposite sides are parallel and equal in length. A trapezium is a quadrilateral which has at least one pair of parallel opposite sides. A convex quadrilateral is Simple, The boundary does not cross itself, star-shaped, The whole interior is visible from a single point, without crossing any edge. De Villiers defines a more generally as any quadrilateral with axes of symmetry through each pair of opposite sides. This definition includes both right-angled rectangles and crossed rectangles, quadrilaterals with two axes of symmetry, each through a pair of opposite sides, belong to the larger class of quadrilaterals with at least one axis of symmetry through a pair of opposite sides. These quadrilaterals comprise isosceles trapezia and crossed isosceles trapezia, a rectangle is cyclic, all corners lie on a single circle. It is equiangular, all its corner angles are equal and it is isogonal or vertex-transitive, all corners lie within the same symmetry orbit. It has two lines of symmetry and rotational symmetry of order 2. The dual polygon of a rectangle is a rhombus, as shown in the table below, the figure formed by joining, in order, the midpoints of the sides of a rectangle is a rhombus and vice versa

6. Rhombus – In Euclidean geometry, a rhombus is a simple quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length, every rhombus is a parallelogram and a kite. A rhombus with right angles is a square, the word rhombus comes from Greek ῥόμβος, meaning something that spins, which derives from the verb ῥέμβω, meaning to turn round and round. The word was used both by Euclid and Archimedes, who used the term solid rhombus for two right circular cones sharing a common base, the surface we refer to as rhombus today is a cross section of this solid rhombus through the apex of each of the two cones. This is a case of the superellipse, with exponent 1. Every rhombus has two diagonals connecting pairs of vertices, and two pairs of parallel sides. Using congruent triangles, one can prove that the rhombus is symmetric across each of these diagonals and it follows that any rhombus has the following properties, Opposite angles of a rhombus have equal measure. The two diagonals of a rhombus are perpendicular, that is, a rhombus is an orthodiagonal quadrilateral, the first property implies that every rhombus is a parallelogram. Thus denoting the common side as a and the diagonals as p and q, not every parallelogram is a rhombus, though any parallelogram with perpendicular diagonals is a rhombus. In general, any quadrilateral with perpendicular diagonals, one of which is a line of symmetry, is a kite, every rhombus is a kite, and any quadrilateral that is both a kite and parallelogram is a rhombus. A rhombus is a tangential quadrilateral and that is, it has an inscribed circle that is tangent to all four sides. As for all parallelograms, the area K of a rhombus is the product of its base, the base is simply any side length a, K = a ⋅ h. The inradius, denoted by r, can be expressed in terms of the p and q as. The dual polygon of a rhombus is a rectangle, A rhombus has all sides equal, a rhombus has opposite angles equal, while a rectangle has opposite sides equal. A rhombus has a circle, while a rectangle has a circumcircle. A rhombus has an axis of symmetry through each pair of opposite vertex angles, the diagonals of a rhombus intersect at equal angles, while the diagonals of a rectangle are equal in length. The figure formed by joining the midpoints of the sides of a rhombus is a rectangle, a rhombohedron is a three-dimensional figure like a cube, except that its six faces are rhombi instead of squares. The rhombic dodecahedron is a polyhedron with 12 congruent rhombi as its faces

Pentagon [videos]
In geometry, a pentagon (from the Greek πέντε pente and γωνία gonia, meaning five and angle) is any five-sided polygon
Image: Bhindi Cut Up
Image: Morning Glory Flower
Image: Sterappel dwarsdrsn
Image: Carambola Starfruit
Hexagon [videos]
In geometry, a hexagon (from Greek ἕξ hex, "six" and γωνία, gonía, "corner, angle") is a six sided polygon or 6-gon.
Giant's Causeway closeup
Image: Graphen
Image: Assembled E ELT mirror segments undergoing testing
Image: Honey comb
Octagon [videos]
In geometry, an octagon (from the Greek ὀκτάγωνον oktágōnon, "eight angles") is an eight-sided polygon or 8-gon. — A
The octagonal floor plan, Dome of the Rock.
Image: Zont 8 ugolnik
Image: Afghancarpet 1
Image: Janggipieces
Dodecagon [videos]
In geometry, a dodecagon or 12-gon is any twelve-sided polygon. — Regular dodecagon — A regular dodecagon is a figure
The Vera Cruz church in Segovia
A 1942 British threepence, reverse
Image: Wooden pattern blocks dodecagon
A regular skew dodecagon seen as zig-zagging edges of a hexagonal antiprism.
Rectangle [videos]
In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as an
Image: Wallpaper group cmm 1
Image: Wallpaper group p 4g 1
A saddle rectangle has 4 nonplanar vertices, alternated from vertices of a cuboid, with a unique minimal surface interior defined as a linear combination of the four vertices, creating a saddle surface. This example shows 4 blue edges of the rectangle, and two green diagonals, all being diagonal of the cuboid rectangular faces.
Image: Crossed rectangles
Rhombus [videos]
In Euclidean geometry, a rhombus(◊) (plural rhombi or rhombuses) is a simple (non-self-intersecting) quadrilateral
Image: Rhombicdodecahedron
Image: Rhombictriacontahedr on
Image: Isohedral tiling p 4 55
Image: Isohedral tiling p 4 51c
Heptagon [videos]
In geometry, a heptagon is a seven-sided polygon or 7-gon. — The heptagon is also occasionally referred to as the
Geometry problem of the surface of a heptagon divided into triangles, on a clay tablet belonging to a school for scribes; Susa, first half of the 2nd millennium BCE
Image: Get the point (301733819)
Symmetries of a regular heptagon. Vertices are colored by their symmetry positions. Blue mirror lines are drawn through vertices and edges. Gyration orders are given in the center.
Image: Neusis heptagon
In mathematics, a hexadecagon (sometimes called a hexakaidecagon) or 16-gon is a sixteen-sided polygon. — Regular hexadecagon...
The hexadecagonal tower from Raphael's ''The Marriage of the Virgin''
A hexadecagrammic pattern from the Alhambra
Image: 8 antiprism skew 16 gon
Trapezoid [videos]
In Euclidean geometry, a convex quadrilateral with at least one pair of parallel sides is referred to as a trapezoid
Example of a trapeziform pronotum outlined on a spurge bug
Trapezoid special cases
Trapezoid
Vertex figure [videos]
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is
truncated cubic honeycomb (partial).
Image: Great icosahedron vertfig
Image: Truncated cubic honeycomb verf
Image: VF truncated cubic
Hendecagon [videos]
In geometry, a hendecagon (also undecagon or endecagon) or 11-gon is an eleven-sided polygon. (The name hendecagon,
Image: Fotothek df tg 0004812 Geometrie ^ Architektur ^ Festungsbau ^ Vermessung
Symmetries of a regular hendecagon. Vertices are colored by their symmetry positions. Blue mirror lines are drawn through vertices and edge. Gyration orders are given in the center.
Image: Regular star polygon 11 2
Image: Regular star polygon 11 3
Triangle [videos]
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with
The Flatiron Building in New York is shaped like a triangular prism
A triangle
Euler diagram of types of triangles, using the definition that isosceles triangles have at least 2 equal sides, i.e. equilateral triangles are isosceles.
Equilateral Triangle
Polygon [videos]
In elementary geometry, a polygon is a plane figure that is bounded by a finite chain of straight line segments
Historical image of polygons (1699)
Some polygons of different kinds: open (excluding its boundary), boundary only (excluding interior), closed (including both boundary and interior), and self-intersecting.
Some different types of polygon
Equilateral triangle [videos]
In geometry, an equilateral triangle is a triangle in which all three sides are equal. In the familiar Euclidean
A regular tetrahedron is made of four equilateral triangles.
Construction of equilateral triangle with compass and straightedge
Image: Equilateral triangle heights
Line segment [videos]
In geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on
historical image – create a line segment (1699)
The geometric definition of a closed line segment: the intersection of all points at or to the right of A with all points at or to the left of B