Finding the Interior Angle of a Regular Polygon

The question asks us to find the measure of a single interior angle of a regular polygon that has 30 sides. A regular polygon is one where all sides are equal in length and all interior angles are equal in measure.

Polygon Angle Formula

To solve this, we can use the formula for the measure of an interior angle of a regular polygon.

Let $n$ be the number of sides of the regular polygon.

The sum of the interior angles of any polygon (regular or irregular) is given by the formula:

Sum of interior angles = \( (n - 2) \times 180^\circ \)

Since the polygon is regular, all its interior angles are equal. Therefore, the measure of one interior angle is the sum of the interior angles divided by the number of sides ($n$):

Measure of one interior angle = \( \frac{(n - 2) \times 180^\circ}{n} \)

Applying the Formula for 30 Sides

In this specific problem, the regular polygon has 30 sides. So, we set $n = 30$.

Now, substitute $n=30$ into the formula:

Measure of one interior angle = \( \frac{(30 - 2) \times 180^\circ}{30} \)

Step-by-Step Calculation

1. Subtract 2 from the number of sides: \( 30 - 2 = 28 \)
2. Multiply the result by \( 180^\circ \): \( 28 \times 180^\circ \)
3. Divide the sum by the number of sides (\(n=30\)): \( \frac{28 \times 180^\circ}{30} \)

Let's simplify the calculation:

Measure of one interior angle = \( \frac{28 \times 180}{30} \)

We can simplify this by dividing 180 by 30:

\( \frac{180}{30} = 6 \)

So, the calculation becomes:

Measure of one interior angle = \( 28 \times 6 \)

Performing the multiplication:

\( 28 \times 6 = 168 \)

Therefore, the measure of one interior angle of a regular polygon with 30 sides is \( 168^\circ \).

Conclusion

The calculation shows that each interior angle of a regular polygon with 30 sides measures \( 168^\circ \). This matches the first option provided.