Question
If each interior angle of a rectangular polygon is 120°.what is the value of each interior angle
Answers
GPT-4o
In this problem, you mentioned a "rectangular polygon" but I believe you meant a "regular polygon" since there is no such thing as a rectangular polygon with angles uniformly distributed. Let's correct it to a "regular polygon," which has all sides and angles equal.
To find the value of each interior angle of a regular polygon with \( n \) sides (where each interior angle is given as 120°), we can use the formula for the interior angle \( \theta \) of a regular polygon:
\[
\theta = \frac{(n-2) \cdot 180^\circ}{n}
\]
We know that:
\[
\theta = 120^\circ
\]
We set the equation equal to 120° and solve for \( n \):
\[
120 = \frac{(n-2) \cdot 180}{n}
\]
To clear the fraction, multiply both sides by \( n \):
\[
120n = (n-2) \cdot 180
\]
Distribute the 180 on the right side:
\[
120n = 180n - 360
\]
Isolate \( n \) by subtracting 180n from both sides:
\[
-60n = -360
\]
Divide both sides by -60:
\[
n = 6
\]
So, the polygon must have 6 sides, meaning it is a regular hexagon. Each interior angle of a regular hexagon is indeed 120°, so the given condition is satisfied. Therefore, each interior angle is correctly 120°.
To find the value of each interior angle of a regular polygon with \( n \) sides (where each interior angle is given as 120°), we can use the formula for the interior angle \( \theta \) of a regular polygon:
\[
\theta = \frac{(n-2) \cdot 180^\circ}{n}
\]
We know that:
\[
\theta = 120^\circ
\]
We set the equation equal to 120° and solve for \( n \):
\[
120 = \frac{(n-2) \cdot 180}{n}
\]
To clear the fraction, multiply both sides by \( n \):
\[
120n = (n-2) \cdot 180
\]
Distribute the 180 on the right side:
\[
120n = 180n - 360
\]
Isolate \( n \) by subtracting 180n from both sides:
\[
-60n = -360
\]
Divide both sides by -60:
\[
n = 6
\]
So, the polygon must have 6 sides, meaning it is a regular hexagon. Each interior angle of a regular hexagon is indeed 120°, so the given condition is satisfied. Therefore, each interior angle is correctly 120°.