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Vertex

Vertex

A corner point of a geometric figure. For a polygon, vertices are where adjacent sides meet. For an angle, the vertex is where the two rays making up the angle meet.

Note: If the figure is a curve or surface, the vertices are the points of maximum curvature.

 

A quadrilateral (tilted square) with each of its four corner points labeled "Vertex".

 

See also

Vertex of a parabola, vertices of an ellipse, vertices of a hyperbola

Example

Problem: A triangle has vertices at A(1, 2), B(5, 2), and C(3, 6). Identify each vertex and count the total number of vertices.
Step 1: A triangle is a polygon with three sides. Each side meets another side at a corner point — that corner point is a vertex.
Step 2: At point A(1, 2), side AB meets side AC. So A is a vertex.
Step 3: At point B(5, 2), side AB meets side BC. So B is a vertex.
Step 4: At point C(3, 6), side AC meets side BC. So C is a vertex.
Step 5: Count the vertices: A, B, and C give a total of 3 vertices. This matches the general rule that an n-sided polygon has n vertices.
Answer: The triangle has 3 vertices: A(1, 2), B(5, 2), and C(3, 6).

Another Example

Problem: Find the vertex of the parabola given by y = 2(x − 3)² + 5.
Step 1: The equation is already in vertex form, y = a(x − h)² + k, where (h, k) is the vertex.
y=a(x−h)2+ky = a(x - h)^2 + k
Step 2: Match the given equation to the form: a = 2, h = 3, k = 5.
y=2(x−3)2+5y = 2(x - 3)^2 + 5
Step 3: The vertex is the point (h, k). Since a = 2 > 0 the parabola opens upward, so the vertex is the lowest point (the minimum).
(3, 5)(3,\, 5)
Answer: The vertex of the parabola is (3, 5).

Frequently Asked Questions

What is the plural of vertex?
The plural of vertex is vertices. For example, a rectangle has four vertices. You may occasionally see 'vertexes' in informal writing, but 'vertices' is the standard mathematical term.
How many vertices does a cube have?
A cube has 8 vertices. Each vertex is a corner where three edges meet. This is consistent with Euler's formula for polyhedra: V − E + F = 2, since a cube has 8 vertices, 12 edges, and 6 faces, giving 8 − 12 + 6 = 2.

Vertex vs. Edge

A vertex is a point — a zero-dimensional location where edges meet. An edge is a line segment — a one-dimensional connection between two vertices. In a triangle, the three corners are vertices while the three line segments connecting them are edges (sides). Every polygon and polyhedron has both vertices and edges, and the two concepts depend on each other: edges are defined by their endpoint vertices, and vertices are defined by the edges that meet there.

Why It Matters

Vertices are fundamental reference points in geometry. When you classify polygons, calculate angles, graph parabolas, or work with 3D shapes, you rely on vertices to anchor measurements and descriptions. In coordinate geometry and computer graphics, shapes are stored and manipulated entirely as lists of vertex coordinates.

Common Mistakes

Mistake: Confusing the vertex of an angle with points on its rays.
Correction: The vertex is only the single point where the two rays originate. Points lying along either ray are not the vertex — they simply lie on the sides of the angle.
Mistake: Thinking the vertex of a parabola is always at the origin.
Correction: The vertex of y = a(x − h)² + k is at (h, k), which can be any point in the plane. It sits at the origin only when h = 0 and k = 0.

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