In classical geometry , a radius ( pl. : radii or radiuses ) of a circle or sphere is any of the line segments from its center to its perimeter , and in more modern usage, it is also their length. The name comes from the Latin radius , meaning ray but also the spoke of a chariot wheel. The typical abbreviation and mathematical variable symbol for radius is R or r . By extension, the diameter D is defined as twice the radius:
37-425: If an object does not have a center, the term may refer to its circumradius , the radius of its circumscribed circle or circumscribed sphere . In either case, the radius may be more than half the diameter, which is usually defined as the maximum distance between any two points of the figure. The inradius of a geometric figure is usually the radius of the largest circle or sphere contained in it. The inner radius of
74-400: 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
111-409: A Cartesian system ) is called the pole , and the ray from the pole in the fixed direction is the polar axis . The distance from the pole is called the radial coordinate or radius , and the angle is the angular coordinate , polar angle , or azimuth . In the cylindrical coordinate system, there is a chosen reference axis and a chosen reference plane perpendicular to that axis. The origin of
148-425: A triangle is a circle that passes through all three vertices . The center of this circle 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
185-426: A reference plane that passes through the origin and is orthogonal to the zenith, and a fixed reference direction in that plane. Circumscribed circle In geometry, a circumscribed circle for a set of points is a circle passing through each of them. Such a circle is said to circumscribe the points or a polygon formed from them; such a polygon is said to be inscribed in
222-407: A ring, tube or other hollow object is the radius of its cavity. For regular polygons , the radius is the same as its circumradius. The inradius of a regular polygon is also called apothem . In graph theory , the radius of a graph is the minimum over all vertices u of the maximum distance from u to any other vertex of the graph. The radius of the circle with perimeter ( circumference ) C
259-439: 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
296-504: Is For many geometric figures, the radius has a well-defined relationship with other measures of the figure. The radius of a circle with area A is The radius of the circle that passes through the three non- collinear points P 1 , P 2 , and P 3 is given by where θ is the angle ∠ P 1 P 2 P 3 . This formula uses the law of sines . If the three points are given by their coordinates ( x 1 , y 1 ) , ( x 2 , y 2 ) , and ( x 3 , y 3 ) ,
333-452: 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
370-415: 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
407-447: 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
SECTION 10
#1732847593159444-403: The radial distance or radius , while the angular coordinate is sometimes referred to as the angular position or as the azimuth . The radius and the azimuth are together called the polar coordinates , as they correspond to a two-dimensional polar coordinate system in the plane through the point, parallel to the reference plane. The third coordinate may be called the height or altitude (if
481-452: 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
518-418: 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
555-407: The circle. Circumcircle , the circumscribed circle of a triangle, which always exists for a given triangle. Cyclic polygon , a general polygon that can be circumscribed by a circle. The vertices of this polygon are concyclic points . All triangles are cyclic polygons. Cyclic quadrilateral , a special case of a cyclic polygon. See also [ edit ] Smallest-circle problem ,
592-402: 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
629-453: 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
666-413: The circumcenter is the point where they cross. Any point on the bisector 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
703-412: 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
740-560: The circumcircle upon which the observer lies. In 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
777-403: 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
SECTION 20
#1732847593159814-418: The circumscribed circle, is called a cyclic polygon , or in the special case n = 4 , 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
851-432: 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
888-399: 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
925-435: The link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Circumscribed_circle&oldid=1224654347 " Category : Set index articles Hidden categories: Articles with short description Short description with empty Wikidata description All set index articles Circumcircle In geometry , the circumscribed circle or circumcircle of
962-500: 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
999-412: The radius can be expressed as The radius r of a regular polygon with n sides of length s is given by r = R n s , where R n = 1 / ( 2 sin π n ) . {\displaystyle R_{n}=1\left/\left(2\sin {\frac {\pi }{n}}\right)\right..} Values of R n for small values of n are given in
1036-409: The reference plane is considered horizontal), longitudinal position , or axial position . In a spherical coordinate system, the radius describes the distance of a point from a fixed origin. Its position if further defined by the polar angle measured between the radial direction and a fixed zenith direction, and the azimuth angle, the angle between the orthogonal projection of the radial direction on
1073-404: The related problem of finding the circle with minimal radius containing an arbitrary set of points, not necessarily passing through them. Inscribed figure [REDACTED] Index of articles associated with the same name This set index article includes a list of related items that share the same name (or similar names). If an internal link incorrectly led you here, you may wish to change
1110-489: 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
1147-417: The system is the point where all three coordinates can be given as zero. This is the intersection between the reference plane and the axis. The axis is variously called the cylindrical or longitudinal axis, to differentiate it from the polar axis , which is the ray that lies in the reference plane, starting at the origin and pointing in the reference direction. The distance from the axis may be called
Radius - Misplaced Pages Continue
1184-412: The table. If s = 1 then these values are also the radii of the corresponding regular polygons. The radius of a d -dimensional hypercube with side s is The polar coordinate system is a two - dimensional coordinate system in which each point on a plane is determined by a distance from a fixed point and an angle from a fixed direction. The fixed point (analogous to the origin of
1221-495: 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
1258-426: 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
1295-443: 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
1332-559: 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
1369-457: The vertices at an angle with the common side, the common angle of departure being 90° minus 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
#158841