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65-400: A regular octagon has Schläfli symbol {8} and can also be constructed as a quasiregular truncated square , t{4}, which alternates two types of edges. A truncated octagon, t{8} is a hexadecagon , {16}. A 3D analog of the octagon can be the rhombicuboctahedron with the triangular faces on it like the replaced edges, if one considers the octagon to be a truncated square. The sum of all

130-400: A x + b y + c z = 0 {\displaystyle ax+by+cz=0} and in barycentric coordinates by x + y + z = 0. {\displaystyle x+y+z=0.} Additionally, the circumcircle of a triangle embedded in three dimensions can be found using a generalized method. Let A , B , C be three-dimensional points, which form the vertices of

195-514: A = 1, this produces the following table: ( Since cot ⁡ x → 1 / x {\displaystyle \cot x\rightarrow 1/x} as x → 0 {\displaystyle x\rightarrow 0} , the area when s = 1 {\displaystyle s=1} tends to n 2 / 4 π {\displaystyle n^{2}/4\pi } as n {\displaystyle n} grows large.) Of all n -gons with

260-634: A Petrie polygon projection plane of the tesseract . The list (sequence A006245 in the OEIS ) defines the number of solutions as eight, by the eight orientations of this one dissection. These squares and rhombs are used in the Ammann–Beenker tilings . A skew octagon is a skew polygon with eight vertices and edges but not existing on the same plane. The interior of such an octagon is not generally defined. A skew zig-zag octagon has vertices alternating between two parallel planes. A regular skew octagon

325-402: A cyclic quadrilateral . All rectangles , isosceles trapezoids , right kites , and regular polygons are cyclic, but not every polygon is. The circumcenter of a triangle can be constructed by drawing any two of the three perpendicular bisectors . For three non-collinear points, these two lines cannot be parallel, and the circumcenter is the point where they cross. Any point on the bisector

390-398: A power of two : The regular octagon can be constructed with meccano bars. Twelve bars of size 4, three bars of size 5 and two bars of size 6 are required. Each side of a regular octagon subtends half a right angle at the centre of the circle which connects its vertices. Its area can thus be computed as the sum of eight isosceles triangles, leading to the result: for an octagon of side

455-541: A . The coordinates for the vertices of a regular octagon centered at the origin and with side length 2 are: Coxeter states that every zonogon (a 2 m -gon whose opposite sides are parallel and of equal length) can be dissected into m ( m -1)/2 parallelograms. In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular octagon , m =4, and it can be divided into 6 rhombs, with one example shown below. This decomposition can be seen as 6 of 24 faces in

520-435: A circle. The value of the internal angle can never become exactly equal to 180°, as the circumference would effectively become a straight line (see apeirogon ). For this reason, a circle is not a polygon with an infinite number of sides. For n > 2, the number of diagonals is 1 2 n ( n − 3 ) {\displaystyle {\tfrac {1}{2}}n(n-3)} ; i.e., 0, 2, 5, 9, ..., for

585-416: A given perimeter, the one with the largest area is regular. Some regular polygons are easy to construct with compass and straightedge ; other regular polygons are not constructible at all. The ancient Greek mathematicians knew how to construct a regular polygon with 3, 4, or 5 sides, and they knew how to construct a regular polygon with double the number of sides of a given regular polygon. This led to

650-636: A number of octagonal churches in Norway . The central space in the Aachen Cathedral , the Carolingian Palatine Chapel , has a regular octagonal floorplan. Uses of octagons in churches also include lesser design elements, such as the octagonal apse of Nidaros Cathedral . Architects such as John Andrews have used octagonal floor layouts in buildings for functionally separating office areas from building services, such as in

715-412: A reference octagon has its eight vertices at the midpoints of the sides of the reference octagon. If squares are constructed all internally or all externally on the sides of the midpoint octagon, then the midpoints of the segments connecting the centers of opposite squares themselves form the vertices of a square. A regular octagon is a closed figure with sides of the same length and internal angles of

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780-404: A regular octagon of side length a is given by In terms of the circumradius R , the area is In terms of the apothem r (see also inscribed figure ), the area is These last two coefficients bracket the value of pi , the area of the unit circle . The area can also be expressed as where S is the span of the octagon, or the second-shortest diagonal; and a is the length of one of

845-447: A side a , the span S is The span, then, is equal to the silver ratio times the side, a. The area is then as above: Expressed in terms of the span, the area is Another simple formula for the area is 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, The two end lengths e on each side (the leg lengths of

910-447: A triangle, square, pentagon, hexagon, ... . The diagonals divide the polygon into 1, 4, 11, 24, ... pieces OEIS :  A007678 . For a regular n -gon inscribed in a unit-radius circle, the product of the distances from a given vertex to all other vertices (including adjacent vertices and vertices connected by a diagonal) equals n . For a regular simple n -gon with circumradius R and distances d i from an arbitrary point in

975-439: A triangle. We start by transposing the system to place C at the origin: The circumradius r is then where θ is the interior angle between a and b . The circumcenter, p 0 , is given by This formula only works in three dimensions as the cross product is not defined in other dimensions, but it can be generalized to the other dimensions by replacing the cross products with following identities: This gives us

1040-461: A uniform antiprism . All edges and internal angles are equal. More generally regular skew polygons can be defined in n -space. Examples include the Petrie polygons , polygonal paths of edges that divide a regular polytope into two halves, and seen as a regular polygon in orthogonal projection. In the infinite limit regular skew polygons become skew apeirogons . A non-convex regular polygon

1105-401: Is dihedral group D n (of order 2 n ): D 2 , D 3 , D 4 , ... It consists of the rotations in C n , together with reflection symmetry in n axes that pass through the center. If n is even then half of these axes pass through two opposite vertices, and the other half through the midpoint of opposite sides. If n is odd then all axes pass through a vertex and the midpoint of

1170-491: Is r16 and no symmetry is labeled a1 . The most common high symmetry octagons are p8 , an isogonal octagon constructed by four mirrors can alternate long and short edges, and d8 , an isotoxal octagon constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are duals of each other and have half the symmetry order of the regular octagon. Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only

1235-618: Is vertex-transitive with equal edge lengths. In three dimensions it is a zig-zag skew octagon and can be seen in the vertices and side edges of a square antiprism with the same D 4d , [2,8] symmetry, order 16. The regular skew octagon is the Petrie polygon for these higher-dimensional regular and uniform polytopes , shown in these skew orthogonal projections of in A 7 , B 4 , and D 5 Coxeter planes . The regular octagon has Dih 8 symmetry, order 16. There are three dihedral subgroups: Dih 4 , Dih 2 , and Dih 1 , and four cyclic subgroups : Z 8 , Z 4 , Z 2 , and Z 1 ,

1300-731: Is 2 nR − ⁠ 1 / 4 ⁠ ns , where s is the side length and R is the circumradius. If d i {\displaystyle d_{i}} are the distances from the vertices of a regular n {\displaystyle n} -gon to any point on its circumcircle, then Coxeter states that every zonogon (a 2 m -gon whose opposite sides are parallel and of equal length) can be dissected into ( m 2 ) {\displaystyle {\tbinom {m}{2}}} or ⁠ 1 / 2 ⁠ m ( m − 1) parallelograms. These tilings are contained as subsets of vertices, edges and faces in orthogonal projections m -cubes . In particular, this

1365-474: Is a regular star polygon . The most common example is the pentagram , which has the same vertices as a pentagon , but connects alternating vertices. For an n -sided star polygon, the Schläfli symbol is modified to indicate the density or "starriness" m of the polygon, as { n / m }. If m is 2, for example, then every second point is joined. If m is 3, then every third point is joined. The boundary of

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1430-471: Is a uniform polyhedron which has just one kind of face. The remaining (non-uniform) convex polyhedra with regular faces are known as the Johnson solids . A polyhedron having regular triangles as faces is called a deltahedron . Circumradius In geometry , the circumscribed circle or circumcircle of a triangle is a circle that passes through all three vertices . The center of this circle

1495-452: Is a unique circle passing through any given three non-collinear points P 1 , P 2 , P 3 . Using Cartesian coordinates to represent these points as spatial vectors , it is possible to use the dot product and cross product to calculate the radius and center of the circle. Let Then the radius of the circle is given by The center of the circle is given by the linear combination where The circumcenter's position depends on

1560-420: Is called the circumcenter of the triangle, and its radius is called the circumradius . The circumcenter is the point of intersection between the three perpendicular bisectors of the triangle's sides, and is a triangle center . More generally, an n -sided polygon with all its vertices on the same circle, also called the circumscribed circle, is called a cyclic polygon , or in the special case n = 4 ,

1625-440: Is customary to drop the prefix regular. For instance, all the faces of uniform polyhedra must be regular and the faces will be described simply as triangle, square, pentagon, etc. For a regular convex n -gon, each interior angle has a measure of: and each exterior angle (i.e., supplementary to the interior angle) has a measure of 360 n {\displaystyle {\tfrac {360}{n}}} degrees, with

1690-432: Is equidistant from the two points that it bisects, from which it follows that this point, on both bisectors, is equidistant from all three triangle vertices. The circumradius is the distance from it to any of the three vertices. An alternative method to determine the circumcenter is to draw any two lines each one departing from one of the vertices at an angle with the common side, the common angle of departure being 90° minus

1755-425: Is the distance from an arbitrary point in the plane to the centroid of a regular n {\displaystyle n} -gon with circumradius R {\displaystyle R} , then where m {\displaystyle m} = 1, 2, …, n − 1 {\displaystyle n-1} . For a regular n -gon, the sum of the perpendicular distances from any interior point to

1820-412: Is true for any regular polygon with an even number of sides, in which case the parallelograms are all rhombi. The list OEIS :  A006245 gives the number of solutions for smaller polygons. The area A of a convex regular n -sided polygon having side s , circumradius R , apothem a , and perimeter p is given by For regular polygons with side s = 1, circumradius R = 1, or apothem

1885-518: The Euclidean plane , it is possible to give explicitly an equation of the circumcircle in terms of the Cartesian coordinates of the vertices of the inscribed triangle. Suppose that are the coordinates of points A, B, C . The circumcircle is then the locus of points v = ( v x , v y ) {\displaystyle \mathbf {v} =(v_{x},v_{y})} in

1950-460: The Gauss–Wantzel theorem . Equivalently, a regular n -gon is constructible if and only if the cosine of its common angle is a constructible number —that is, can be written in terms of the four basic arithmetic operations and the extraction of square roots. A regular skew polygon in 3-space can be seen as nonplanar paths zig-zagging between two parallel planes, defined as the side-edges of

2015-563: The Intelsat Headquarters of Washington or Callam Offices in Canberra. The octagon , as a truncated square , is first in a sequence of truncated hypercubes : As an expanded square, it is also first in a sequence of expanded hypercubes: Regular polygon In Euclidean geometry , a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have

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2080-415: The circumsphere of a tetrahedron . A unit vector perpendicular to the plane containing the circle is given by Hence, given the radius, r , center, P c , a point on the circle, P 0 and a unit normal of the plane containing the circle, ⁠ n ^ , {\displaystyle {\widehat {n}},} ⁠ one parametric equation of the circle starting from

2145-750: The g8 subgroup has no degrees of freedom but can be seen as directed edges . The octagonal shape is used as a design element in architecture. The Dome of the Rock has a characteristic octagonal plan. The Tower of the Winds in Athens is another example of an octagonal structure. The octagonal plan has also been in church architecture such as St. George's Cathedral, Addis Ababa , Basilica of San Vitale (in Ravenna, Italia), Castel del Monte (Apulia, Italia), Florence Baptistery , Zum Friedefürsten Church (Germany) and

2210-447: The locus of zeros of the determinant of this matrix: Using cofactor expansion , let we then have a | v | 2 − 2 S v − b = 0 {\displaystyle a|\mathbf {v} |^{2}-2\mathbf {Sv} -b=0} where S = ( S x , S y ) , {\displaystyle \mathbf {S} =(S_{x},S_{y}),} and – assuming

2275-476: The n sides is n times the apothem (the apothem being the distance from the center to any side). This is a generalization of Viviani's theorem for the n = 3 case. The circumradius R from the center of a regular polygon to one of the vertices is related to the side length s or to the apothem a by For constructible polygons , algebraic expressions for these relationships exist (see Bicentric polygon § Regular polygons ) . The sum of

2340-457: The Cartesian plane satisfying the equations guaranteeing that the points A , B , C , v are all the same distance r from the common center u {\displaystyle \mathbf {u} } of the circle. Using the polarization identity , these equations reduce to the condition that the matrix has a nonzero kernel . Thus the circumcircle may alternatively be described as

2405-418: The actual circumcenter of △ ABC follows as The circumcenter has trilinear coordinates where α, β, γ are the angles of the triangle. In terms of the side lengths a, b, c , the trilinears are The circumcenter has barycentric coordinates where a, b, c are edge lengths BC , CA , AB respectively) of the triangle. In terms of the triangle's angles α, β, γ , the barycentric coordinates of

2470-416: The angle of the opposite vertex. (In the case of the opposite angle being obtuse, drawing a line at a negative angle means going outside the triangle.) In coastal navigation , a triangle's circumcircle is sometimes used as a way of obtaining a position line using a sextant when no compass is available. The horizontal angle between two landmarks defines the circumcircle upon which the observer lies. In

2535-402: The circumcenter S a {\displaystyle {\tfrac {\mathbf {S} }{a}}} and the circumradius b a + | S | 2 a 2 . {\displaystyle {\sqrt {{\tfrac {b}{a}}+{\tfrac {|\mathbf {S} |^{2}}{a^{2}}}}}.} A similar approach allows one to deduce the equation of

2600-453: The circumcenter are Since the Cartesian coordinates of any point are a weighted average of those of the vertices, with the weights being the point's barycentric coordinates normalized to sum to unity, the circumcenter vector can be written as Here U is the vector of the circumcenter and A, B, C are the vertex vectors. The divisor here equals 16 S where S is the area of the triangle. As stated previously In Euclidean space , there

2665-412: The circumcircle in barycentric coordinates x  : y  : z is a 2 x + b 2 y + c 2 z = 0. {\displaystyle {\tfrac {a^{2}}{x}}+{\tfrac {b^{2}}{y}}+{\tfrac {c^{2}}{z}}=0.} The isogonal conjugate of the circumcircle is the line at infinity, given in trilinear coordinates by

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2730-403: The circumcircle, called the circumdiameter and equal to twice the circumradius , can be computed as the length of any side of the triangle divided by the sine of the opposite angle : As a consequence of the law of sines , it does not matter which side and opposite angle are taken: the result will be the same. The diameter of the circumcircle can also be expressed as where a, b, c are

2795-399: The circumradius as The regular octagon, in terms of the side length a , has three different types of diagonals : The formula for each of them follows from the basic principles of geometry. Here are the formulas for their length: A regular octagon at a given circumcircle may be constructed as follows: A regular octagon can be constructed using a straightedge and a compass , as 8 = 2,

2860-401: The constructibility of regular polygons: (A Fermat prime is a prime number of the form 2 ( 2 n ) + 1. {\displaystyle 2^{\left(2^{n}\right)}+1.} ) Gauss stated without proof that this condition was also necessary , but never published his proof. A full proof of necessity was given by Pierre Wantzel in 1837. The result is known as

2925-432: The following equation for the circumradius r : and the following equation for the circumcenter p 0 : which can be simplified to: The Cartesian coordinates of the circumcenter U = ( U x , U y ) {\displaystyle U=\left(U_{x},U_{y}\right)} are with Without loss of generality this can be expressed in a simplified form after translation of

2990-445: The internal angles of any octagon is 1080°. As with all polygons, the external angles total 360°. If squares are constructed all internally or all externally on the sides of an octagon, then the midpoints of the segments connecting the centers of opposite squares form a quadrilateral that is both equidiagonal and orthodiagonal (that is, whose diagonals are equal in length and at right angles to each other). The midpoint octagon of

3055-401: The last implying no symmetry. On the regular octagon, there are eleven distinct symmetries. John Conway labels full symmetry as r16 . The dihedral symmetries are divided depending on whether they pass through vertices ( d for diagonal) or edges ( p for perpendiculars) Cyclic symmetries in the middle column are labeled as g for their central gyration orders. Full symmetry of the regular form

3120-399: The lengths of the sides of the triangle and s = a + b + c 2 {\displaystyle s={\tfrac {a+b+c}{2}}} is the semiperimeter. The expression s ( s − a ) ( s − b ) ( s − c ) {\displaystyle \scriptstyle {\sqrt {s(s-a)(s-b)(s-c)}}} above

3185-407: The opposite side. All regular simple polygons (a simple polygon is one that does not intersect itself anywhere) are convex. Those having the same number of sides are also similar . An n -sided convex regular polygon is denoted by its Schläfli symbol { n }. For n < 3, we have two degenerate cases: In certain contexts all the polygons considered will be regular. In such circumstances it

3250-404: The perpendiculars from a regular n -gon's vertices to any line tangent to the circumcircle equals n times the circumradius. The sum of the squared distances from the vertices of a regular n -gon to any point on its circumcircle equals 2 nR where R is the circumradius. The sum of the squared distances from the midpoints of the sides of a regular n -gon to any point on the circumcircle

3315-441: The plane to the vertices, we have For higher powers of distances d i {\displaystyle d_{i}} from an arbitrary point in the plane to the vertices of a regular n {\displaystyle n} -gon, if then and where m {\displaystyle m} is a positive integer less than n {\displaystyle n} . If L {\displaystyle L}

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3380-500: The point P 0 and proceeding in a positively oriented (i.e., right-handed ) sense about ⁠ n ^ {\displaystyle {\widehat {n}}} ⁠ is the following: An equation for the circumcircle in trilinear coordinates x  : y  : z is a x + b y + c z = 0. {\displaystyle {\tfrac {a}{x}}+{\tfrac {b}{y}}+{\tfrac {c}{z}}=0.} An equation for

3445-573: The polygon winds around the center m times. The (non-degenerate) regular stars of up to 12 sides are: m and n must be coprime , or the figure will degenerate. The degenerate regular stars of up to 12 sides are: Depending on the precise derivation of the Schläfli symbol, opinions differ as to the nature of the degenerate figure. For example, {6/2} may be treated in either of two ways: All regular polygons are self-dual to congruency, and for odd n they are self-dual to identity. In addition,

3510-423: The question being posed: is it possible to construct all regular n -gons with compass and straightedge? If not, which n -gons are constructible and which are not? Carl Friedrich Gauss proved the constructibility of the regular 17-gon in 1796. Five years later, he developed the theory of Gaussian periods in his Disquisitiones Arithmeticae . This theory allowed him to formulate a sufficient condition for

3575-417: The regular star figures (compounds), being composed of regular polygons, are also self-dual. A uniform polyhedron has regular polygons as faces, such that for every two vertices there is an isometry mapping one into the other (just as there is for a regular polygon). A quasiregular polyhedron is a uniform polyhedron which has just two kinds of face alternating around each vertex. A regular polyhedron

3640-446: The same length). Regular polygons may be either convex , star or skew . In the limit , a sequence of regular polygons with an increasing number of sides approximates a circle , if the perimeter or area is fixed, or a regular apeirogon (effectively a straight line ), if the edge length is fixed. These properties apply to all regular polygons, whether convex or star : The symmetry group of an n -sided regular polygon

3705-501: The same size. It has eight lines of reflective symmetry and rotational symmetry of order 8. A regular octagon is represented by the Schläfli symbol {8}. The internal angle at each vertex of a regular octagon is 135 ° ( 3 π 4 {\displaystyle \scriptstyle {\frac {3\pi }{4}}} radians ). The central angle is 45° ( π 4 {\displaystyle \scriptstyle {\frac {\pi }{4}}} radians). The area of

3770-489: The sides of the triangle coincide with angles at which sides meet each other. The side opposite angle α meets the circle twice: once at each end; in each case at angle α (similarly for the other two angles). This is due to the alternate segment theorem , which states that the angle between the tangent and chord equals the angle in the alternate segment. In this section, the vertex angles are labeled A, B, C and all coordinates are trilinear coordinates : The diameter of

3835-403: The sides, or bases. This is easily proven if one takes an octagon, draws a square around the outside (making sure that four of the eight sides overlap with the four sides of the square) and then takes the corner triangles (these are 45–45–90 triangles ) and places them with right angles pointed inward, forming a square. The edges of this square are each the length of the base. Given the length of

3900-420: The sum of the exterior angles equal to 360 degrees or 2π radians or one full turn. As n approaches infinity, the internal angle approaches 180 degrees. For a regular polygon with 10,000 sides (a myriagon ) the internal angle is 179.964°. As the number of sides increases, the internal angle can come very close to 180°, and the shape of the polygon approaches that of a circle. However the polygon can never become

3965-495: The three points were not in a line (otherwise the circumcircle is that line that can also be seen as a generalized circle with S at infinity) – | v − S a | 2 = b a + | S | 2 a 2 , {\displaystyle \left|\mathbf {v} -{\tfrac {\mathbf {S} }{a}}\right|^{2}={\tfrac {b}{a}}+{\tfrac {|\mathbf {S} |^{2}}{a^{2}}},} giving

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4030-429: The triangles (green in the image) truncated from the square), as well as being e = a / 2 , {\displaystyle e=a/{\sqrt {2}},} may be calculated as The circumradius of the regular octagon in terms of the side length a is and the inradius is (that is one-half the silver ratio times the side, a , or one-half the span, S ) The inradius can be calculated from

4095-426: The type of triangle: These locational features can be seen by considering the trilinear or barycentric coordinates given above for the circumcenter: all three coordinates are positive for any interior point, at least one coordinate is negative for any exterior point, and one coordinate is zero and two are positive for a non-vertex point on a side of the triangle. The angles which the circumscribed circle forms with

4160-443: The vectors from vertex A' to these vertices. Observe that this trivial translation is possible for all triangles and the circumcenter U ′ = ( U x ′ , U y ′ ) {\displaystyle U'=(U'_{x},U'_{y})} of the triangle △ A'B'C' follow as with Due to the translation of vertex A to the origin, the circumradius r can be computed as and

4225-559: The vertex A to the origin of the Cartesian coordinate systems, i.e., when A ′ = A − A = ( A x ′ , A y ′ ) = ( 0 , 0 ) . {\displaystyle A'=A-A=(A'_{x},A'_{y})=(0,0).} In this case, the coordinates of the vertices B ′ = B − A {\displaystyle B'=B-A} and C ′ = C − A {\displaystyle C'=C-A} represent

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