What is the measure of one of the interior angle of a regular dodecagon?
A. 150°
B. 135°
C. 108°
D. 144°
The correct answer and explanation is:
The correct answer is D. 144°.
To calculate the measure of one of the interior angles of a regular dodecagon (12-sided polygon), we can use the formula for the interior angle of a regular polygon: Interior angle=(n−2)×180°n\text{Interior angle} = \frac{(n – 2) \times 180°}{n}
where nn is the number of sides of the polygon. In this case, a dodecagon has n=12n = 12 sides. Substituting this value into the formula: Interior angle=(12−2)×180°12\text{Interior angle} = \frac{(12 – 2) \times 180°}{12} Interior angle=10×180°12\text{Interior angle} = \frac{10 \times 180°}{12} Interior angle=1800°12\text{Interior angle} = \frac{1800°}{12} Interior angle=150°\text{Interior angle} = 150°
So, the interior angle of a regular dodecagon is 150°.
If the question were asking for the exterior angle, we would use the formula: Exterior angle=360°n\text{Exterior angle} = \frac{360°}{n}
For a dodecagon, this would be: Exterior angle=360°12=30°\text{Exterior angle} = \frac{360°}{12} = 30°
Finally, the interior angle and the exterior angle of a regular polygon are supplementary, meaning they add up to 180°. So, for a dodecagon: Interior angle+Exterior angle=180°\text{Interior angle} + \text{Exterior angle} = 180° 150°+30°=180°150° + 30° = 180°
Therefore, 150° is the correct measure of one of the interior angles.