How to find each interior angle of a polygon?
Understand the Problem
The question is asking how to calculate the measure of each interior angle of a polygon, which involves using the formula for the interior angles depending on the number of sides the polygon has.
Answer
The measure of each interior angle of a regular polygon is given by the formula $ \frac{(n - 2) \times 180^\circ}{n} $.
Answer for screen readers
The measure of each interior angle of a regular polygon is given by the formula:
$$ \frac{(n - 2) \times 180^\circ}{n} $$
Steps to Solve
- Identify the number of sides of the polygon
To calculate the measure of each interior angle, we first need to know how many sides the polygon has. Let’s denote the number of sides as $n$.
- Use the formula for sum of interior angles
The sum of the interior angles of a polygon can be calculated using the formula:
$$ S = (n - 2) \times 180^\circ $$
where $S$ is the sum of the interior angles and $n$ is the number of sides.
- Calculate the measure of each interior angle
To find the measure of each interior angle in a regular polygon (where all angles are equal), divide the sum of the interior angles by the number of sides:
$$ \text{Measure of each interior angle} = \frac{S}{n} = \frac{(n - 2) \times 180^\circ}{n} $$
The measure of each interior angle of a regular polygon is given by the formula:
$$ \frac{(n - 2) \times 180^\circ}{n} $$
More Information
This formula shows that as the number of sides $n$ increases, the measure of each interior angle approaches $180^\circ$, which is the angle in a straight line. For example, a triangle ($n = 3$) has angles of $60^\circ$, while a regular hexagon ($n = 6$) has angles of $120^\circ$.
Tips
- A common mistake is forgetting to properly substitute the number of sides $n$ into the formulas.
- Confusing the formulas for the sum of exterior angles and interior angles; remember that the sum of exterior angles of any polygon is always $360^\circ$.
