Euler’s characteristic χ for a tetrahedron

For a convex polyhedron (topologically a sphere), Euler’s relation says χ = V − E + F = 2. Below we draw a tetrahedron, count its vertices V, edges E, and faces F, and verify the identity.

What this is

A regular tetrahedron (the simplest polyhedron). It has four vertices, six edges, and four triangular faces. We include dashed strokes for edges that are visually “behind” the solid.

Answer — drawing

A B C D Dashed = visually hidden edge

Counts

Vertices V

—

Edges E

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Faces F

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Euler χ

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χ = V − E + F = ?

The counts are computed directly from the SVG elements (.vertex, .edge, .face).

Reason

  1. Vertices. A tetrahedron has 4 vertices.
  2. Edges. Every pair of vertices is connected (graph K₄): choose 2 from 4 gives E = C(4,2) = 6.
  3. Faces. Each triangular face is determined by 3 vertices: F = C(4,3) = 4. Equivalently, use 3F = 2E (each of 4 faces has 3 edges, each edge borders 2 faces): F = 2E/3 = 4.
  4. Therefore χ = V − E + F = 4 − 6 + 4 = 2.

Check

This page programmatically reads the SVG and verifies the identity. You can inspect the DOM: there are 4 <circle class="vertex"> nodes, 6 <line class="edge"> nodes, and 4 <polygon class="face"> nodes.