A triangle, with interior vertex angles a, b, and c along with exterior angle d
In geometry, a vertex angle is an angle (shape) associated with a vertex of an n-dimensional polytope. In two dimensions it refers to the angle formed by two intersecting lines, such as at a "corner" (vertex) of a polygon.[1] In higher dimensions there can be more than two lines (edges) meeting at a vertex, making a description of the angle shape more complicated.
In three dimensions and three-dimensional polyhedra, a vertex angle is a polyhedralangle or n-hedral angle.[2] It is described by a sequence of nface angles, which are the angles formed by two edges of the polyhedron meeting at the vertex, or by a sequence of n dihedral angles, which are the angles between two faces sharing the vertex. The angle may be quantified using a single number by the interior solid angle at the vertex (the spherical excess), which is related to the sum of the dihedral angles, or by the angular defect (or excess) of the vertex, which is related to the sum of the face angles, or other metrics such as the polar sine. The simplest type of polyhedral angle is a trihedral angle or trihedron (bounded by three planes), as found at the vertices of a parallelepiped or tetrahedron.[3][4]
For higher-dimensional polytopes, a vertex angle can be quantified using a higher-dimensional solid angle, i.e. by the portion of the n-sphere around the vertex that is interior to the polytope. Face and dihedral angles also generalize to higher dimensions.[citation needed]
The term vertex angle is sometimes used synonymously with face angle, i.e. the angle between two edges meeting at a vertex. It may also refer to the (higher-dimensional) interior solid angle at a vertex.
geometry, a vertexangle is an angle (shape) associated with a vertex of an n-dimensional polytope. In two dimensions it refers to the angle formed by two...
geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles formed...
may speak of the angle subtended by an arc of a circle when the angle'svertex is the centre of the circle. Central angle Inscribed angle Definition of subtended...
polygon has exactly one internal angle per vertex. If every internal angle of a simple polygon is less than a straight angle (π radians or 180°), then the...
Define the vertexangle θa to be the angle BOC and define θb, θc correspondingly. Let ϕ a b {\displaystyle \phi _{ab}} be the dihedral angle between the...
opposite vertexangles, while a rectangle has an axis of symmetry through each pair of opposite sides. The diagonals of a rhombus intersect at equal angles, while...
equal sides A rhombus with a right vertexangle A rhombus with all angles equal A parallelogram with one right vertexangle and two adjacent equal sides A...
angle bisector of the vertexangle of an isosceles triangle will also bisect the opposite side. There exist many different ways of proving the angle bisector...
length and the opposite angles of a parallelogram are of equal measure. The congruence of opposite sides and opposite angles is a direct consequence of...
corresponding side sequence vwxyz we have vertexangle a appearing between sides a and b then its corresponding vertexangle v must appear between sides v and...
semicircle (with a vertex on the semicircle and its defining rays going through the endpoints of the semicircle) is a right angle. Two application examples...
A central angle is an angle whose apex (vertex) is the center O of a circle and whose legs (sides) are radii intersecting the circle in two distinct points...
(identity; 1) rotation about an axis through a vertex, perpendicular to the opposite plane, by an angle of ±120°: 4 axes, 2 per axis, together 8 ((1 2...
In fiber optics, the radiation angle is half the vertexangle of the cone of light emitted at the exit face of an optical fiber. The cone boundary is usually...
point V. Draw an angle whose vertex is point V and whose sides pass through points A, B. Draw line OA. Angle ∠BOA is a central angle; call it θ. Lines...
is acute. An angle bisector of a triangle is a straight line through a vertex which cuts the corresponding angle in half. The three angle bisectors intersect...
at the right-angledvertex, the right triangle's orthocenter—the intersection of its three altitudes—coincides with the right-angledvertex. The radius...
a vertex configuration is a shorthand notation for representing the vertex figure of a polyhedron or tiling as the sequence of faces around a vertex. For...
right angles, or a half-turn). A triangle has three angles, one at each vertex, bounded by a pair of adjacent sides. It was unknown for a long time whether...
size) regular polygons (all angles congruent and all edges congruent), and the same number of faces meet at each vertex. There are only five such polyhedra:...
solved by means of the right angle hook." The construction begins with drawing a circle passing through the vertex P of the angle to be trisected, centered...
In geometry, the polar sine generalizes the sine function of angle to the vertexangle of a polytope. It is denoted by psin. Let v1, ..., vn (n ≥ 1) be...
angle included by the legs is called the vertexangle and the angles that have the base as one of their sides are called the base angles. The vertex opposite...
"pseudo-rhombicuboctahedron". The number of vertices is 720° divided by the vertexangle defect. The cuboctahedron and icosidodecahedron are edge-uniform and...
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