By Jon Zamboni Mar 12, 2023 12:15 am EST
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In geometry, a vertex is the point where two or more edges meet, forming a corner. Every shape—whether two‑ or three‑dimensional—has vertices. For example, a square has four vertices, one at each corner. A vertex also denotes the tip of an angle or the turning point on a graph. The term derives from Latin, meaning “crown.”
A vertex (plural: vertices) is the point where two straight lines or edges intersect.
When two line segments intersect, the meeting point is called a vertex, regardless of whether the lines cross or form a corner. Because angles are defined by two rays that share a common endpoint, that shared endpoint is also a vertex.
In a 2‑D shape, edges form the boundary, and each junction of two edges is a vertex. A triangle has three edges and three vertices; a quadrilateral has four, and so on. Polygons—shapes with at least three sides—always have the same number of vertices as sides. The sum of interior angles for any polygon is calculated as:
Sum of Angles = (Number of sides – 2) × 180°.
Not every flat shape has vertices. Circles and ovals consist of a single continuous edge with no corners, so they contain no vertices. A semicircle also lacks vertices because its edge is a combination of a curved line and a straight line, not two straight lines.
3‑D objects have faces, edges, and vertices. For instance, a cube has six square faces, twelve straight edges, and eight vertices where three edges meet. Each corner of the cube is a vertex. Some 3‑D figures, like spheres, have no vertices because they lack intersecting edges.
Euler’s theorem provides a relationship among faces (F), vertices (V), and edges (E) for any polyhedron:
V + F – E = 2.
Examples: an octahedron has 8 faces, 6 vertices, and 12 edges; a tetrahedron has 4 triangular faces, 4 vertices, and 6 edges. The formula applies to prisms, cuboids, and any solid composed of straight edges.
In algebra, the vertex is the turning point on the graph of a quadratic function. A parabola, described by y = ax² + bx + c, opens upward if a > 0 and downward if a < 0. Its vertex lies at the minimum (or maximum) point: for y = x², the vertex is (0, 0).
The x‑coordinate of a parabola’s vertex can be found with:
x_vertex = –b / (2a).
