A regular polygon has 15 sides. Find the size of each interior angle.
The correct answer and explanation is:
To find the size of each interior angle of a regular polygon with 15 sides, you can use the formula: Interior angle=(n−2)×180∘n\text{Interior angle} = \frac{(n – 2) \times 180^\circ}{n}
where nn is the number of sides of the polygon.
In this case, the polygon has 15 sides, so: Interior angle=(15−2)×180∘15\text{Interior angle} = \frac{(15 – 2) \times 180^\circ}{15} Interior angle=13×180∘15\text{Interior angle} = \frac{13 \times 180^\circ}{15} Interior angle=2340∘15\text{Interior angle} = \frac{2340^\circ}{15} Interior angle=156∘\text{Interior angle} = 156^\circ
So, each interior angle of a regular polygon with 15 sides measures 156°.
Explanation:
The formula for finding the interior angle of a regular polygon is derived from the total sum of the interior angles. For any polygon with nn sides, the total sum of the interior angles can be calculated by the formula: Sum of interior angles=(n−2)×180∘\text{Sum of interior angles} = (n – 2) \times 180^\circ
In this case, the total sum of the interior angles for a 15-sided polygon is: (15−2)×180∘=13×180∘=2340∘(15 – 2) \times 180^\circ = 13 \times 180^\circ = 2340^\circ
Since the polygon is regular, all the interior angles are equal. To find the measure of one interior angle, divide the total sum of the interior angles by the number of sides: 2340∘15=156∘\frac{2340^\circ}{15} = 156^\circ
Therefore, each interior angle of a regular polygon with 15 sides is 156°. This method can be applied to any regular polygon by substituting the appropriate value for nn, the number of sides.