# Topology : Euler Characteristic

## Example Questions

### Example Question #1 : Euler Characteristic

Calculate the Euler Characteristic given the following Betti numbers.

None of the answers are correct.

Explanation:

The Euler Characteristic is calculated by the following equation.

This is also written as,

Recall that the Betti numbers represent the vertices, edges, and faces of the object.

In this particular question the Betti numbers are known therefore, substitute them into the formula to calculate the Euler Characteristic and solve.

### Example Question #2 : Euler Characteristic

What is the Euler Characteristic associated with a 2-dimensional square?

None of the answers are correct.

Explanation:

The Euler Characteristic is calculated by the following equation.

This is also written as,

Recall that the Betti numbers represent the vertices, edges, and faces of the object.

In this particular question the Betti numbers can be calculated therefore,

now substitute them into the formula to calculate the Euler Characteristic and solve.

### Example Question #3 : Euler Characteristic

What is the Euler Characteristic for a line.

A line doesn't have an Euler Characteristic.

Explanation:

The Euler Characteristic is calculated by the following equation.

This is also written as,

Recall that the Betti numbers represent the vertices, edges, and faces of the object.

In this particular question the Betti numbers can be calculated therefore,

now substitute them into the formula to calculate the Euler Characteristic and solve.

### Example Question #4 : Euler Characteristic

What is the Euler Characteristic associated with a 3-dimensional cube?

Explanation:

The Euler Characteristic is calculated by the following equation.

This is also written as,

Recall that the Betti numbers represent the vertices, edges, and faces of the object.

In this particular question the Betti numbers can be calculated therefore,

now substitute them into the formula to calculate the Euler Characteristic and solve.

### Example Question #5 : Euler Characteristic

What is the Euler Characteristic for a point?

A point doesn't have an Euler Characteristic.

Explanation:

The Euler Characteristic is calculated by the following equation.

This is also written as,

Recall that the Betti numbers represent the vertices, edges, and faces of the object.

In this particular question the Betti numbers can be calculated therefore,

now substitute them into the formula to calculate the Euler Characteristic and solve.

### Example Question #6 : Euler Characteristic

What is the Euler Characteristic for a triangle?

Explanation:

The Euler Characteristic is calculated by the following equation.

This is also written as,

Recall that the Betti numbers represent the vertices, edges, and faces of the object.

In this particular question the Betti numbers can be calculated therefore,

now substitute them into the formula to calculate the Euler Characteristic and solve.