Question

If an interior angle of a regular polygon measures 144°, how many sides does the polygon have?

Answer 1

To determine the number of sides in a regular polygon given that the measure of each interior angle is \(144^\circ\), we can use the formula for the measure of an interior angle of a regular polygon:

\[
\text{Interior angle} = \frac{(n-2) \times 180^\circ}{n}
\]

where \(n\) is the number of sides of the polygon.

Given that the interior angle is \(144^\circ\), we set up the equation:

\[
\frac{(n-2) \times 180}{n} = 144
\]

First, we multiply both sides of the equation by \(n\) to eliminate the fraction:

\[
(n - 2) \times 180 = 144n
\]

Next, we expand the left side:

\[
180n - 360 = 144n
\]

Now, we subtract \(144n\) from both sides to combine like terms:

\[
180n - 144n - 360 = 0
\]

This simplifies to:

\[
36n - 360 = 0
\]

Next, we add \(360\) to both sides:

\[
36n = 360
\]

Now, divide both sides by \(36\):

\[
n = \frac{360}{36} = 10
\]

Thus, the number of sides in the polygon is \( \boxed{10} \).

To verify, we can calculate the interior angle for a 10-sided polygon (decagon):

1. Calculate the total interior angles using the formula \( (n-2) \times 180 \):

\[
(10 - 2) \times 180 = 8 \times 180 = 1440^\circ
\]

2. Calculate the measure of each interior angle:

\[
\frac{1440}{10} = 144^\circ
\]

This confirms that our calculation was correct, and the polygon is indeed a decagon with \( \boxed{10} \) sides.