The sum of all interior angles of a regular polygon is `1080^(@)`. What is the measure of each of its interior angles ?

Options

A. `135^(@)`
B. `120^(@)`
C. `156^(@)`
D. `144^(@)`

Answer

To solve the problem of finding the measure of each interior angle of a regular polygon with a sum of interior angles equal to \(1080^\circ\), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the formula for the sum of interior angles**:
The sum of the interior angles \(S\) of a polygon with \(n\) sides is given by the formula:
\[ S = 180(n - 2) \]

2. **Set up the equation**:
Since we know the sum of the interior angles is \(1080^\circ\), we can set up the equation:
\[ 180(n - 2) = 1080 \]

3. **Solve for \(n\)**:
To find \(n\), we first divide both sides of the equation by \(180\):
\[ n - 2 = \frac{1080}{180} \]
Simplifying the right side gives:
\[ n - 2 = 6 \]
Now, add \(2\) to both sides:
\[ n = 6 + 2 = 8 \]

4. **Determine the number of sides**:
The polygon has \(n = 8\) sides, which means it is an octagon.

5. **Calculate the measure of each interior angle**:
The measure of each interior angle \(p\) of a regular polygon can be calculated using the formula:
\[ p = \frac{S}{n} \]
Substituting the known values:
\[ p = \frac{1080}{8} \]
Performing the division:
\[ p = 135^\circ \]

6. **Final answer**:
Therefore, the measure of each interior angle of the polygon is \(135^\circ\).