Euler Characteristic Calculator
Compute the Euler characteristic from vertices, edges, and faces to classify polyhedra, genus, and surface topology.
What Is the Euler Characteristic?
The Euler characteristic $\chi$ is a topological invariant that summarizes the shape of a polyhedron or surface using only counts of vertices, edges, and faces.
Euler Formula
$$\chi = V - E + F$$For a closed orientable surface, genus $g$ and Euler characteristic satisfy $\chi = 2 - 2g$. A sphere has $g = 0$ and $\chi = 2$. A torus has $g = 1$ and $\chi = 0$.
Platonic Solids
All five Platonic solids are topologically equivalent to a sphere, so each has $\chi = 2$ even though $V$, $E$, and $F$ differ. For example, a cube has $8 - 12 + 6 = 2$.
Related tools: explore the Volume Calculator for solid geometry measurements and the Surface Area Calculator for face-based geometry.
Frequently Asked Questions
Why is the Euler characteristic important?
It stays unchanged under continuous deformations that do not tear or glue the surface. That makes it a powerful tool for classifying shapes in topology.
What does chi = 0 mean?
For a closed orientable surface, $\chi = 0$ means genus 1, which corresponds to a torus (one hole).
Do all polyhedra have positive Euler characteristic?
Closed orientable polyhedra homeomorphic to a sphere have $\chi = 2$. Surfaces with holes can have lower values such as 0, -2, or -4.
How do I count edges correctly?
Each edge is shared by exactly two faces in a closed convex polyhedron. If you count edges face by face, divide by two to avoid double counting.
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