A regular polygon has 16 sides.
a. What is the sum of the interior angles of the 16-gon?
b. What is the measure of an interior angle of the 16-gon?
c. What is the measure of an exterior angle of the 16-gon?
The correct answer and explanation is :
Let’s break down the questions one by one.
a. Sum of the interior angles of a 16-gon
To calculate the sum of the interior angles of a regular polygon, we use the formula:
$$
\text{Sum of interior angles} = (n – 2) \times 180^\circ
$$
where $n$ is the number of sides of the polygon. For a 16-gon, $n = 16$, so:
$$
\text{Sum of interior angles} = (16 – 2) \times 180^\circ = 14 \times 180^\circ = 2520^\circ
$$
Thus, the sum of the interior angles of a 16-gon is 2520 degrees.
b. Measure of an interior angle of a 16-gon
For a regular polygon, all the interior angles are congruent. To find the measure of one interior angle, divide the sum of the interior angles by the number of sides $n$:
$$
\text{Measure of an interior angle} = \frac{\text{Sum of interior angles}}{n} = \frac{2520^\circ}{16} = 157.5^\circ
$$
So, the measure of an interior angle of the 16-gon is 157.5 degrees.
c. Measure of an exterior angle of a 16-gon
The exterior angles of a polygon are related to the interior angles by the fact that each pair of interior and exterior angles are supplementary (they add up to 180 degrees). For a regular polygon, the measure of each exterior angle can be calculated using the formula:
$$
\text{Exterior angle} = \frac{360^\circ}{n}
$$
For a 16-gon, $n = 16$, so:
$$
\text{Exterior angle} = \frac{360^\circ}{16} = 22.5^\circ
$$
Thus, the measure of an exterior angle of the 16-gon is 22.5 degrees.
Explanation:
- The sum of the interior angles of any polygon with $n$ sides can be found using the formula $(n – 2) \times 180^\circ$. This is derived from the fact that a polygon can be divided into triangles, and each triangle has interior angles summing to 180°. For a polygon with $n$ sides, the number of triangles is $n – 2$, hence the formula.
- The measure of each interior angle of a regular polygon is the total sum of interior angles divided by the number of sides, because all the interior angles are congruent in a regular polygon.
- The exterior angle of a regular polygon is found by dividing 360° (the total angle around a point) by the number of sides of the polygon. Since exterior angles and interior angles are supplementary, this approach gives the correct angle measurement for exterior angles in a regular polygon.