To the nearest degree what is the measure of each exterior angle of a regular dodecagon
The Correct Answer and Explanation is:
To find the measure of each exterior angle of a regular dodecagon, we use the formula for the exterior angle of a regular polygon:Exterior Angle=360∘n\text{Exterior Angle} = \frac{360^\circ}{n}Exterior Angle=n360∘
where nnn is the number of sides of the polygon. For a dodecagon, n=12n = 12n=12, because a dodecagon has 12 sides.
So, the exterior angle of a regular dodecagon is:Exterior Angle=360∘12=30∘\text{Exterior Angle} = \frac{360^\circ}{12} = 30^\circExterior Angle=12360∘=30∘
Thus, each exterior angle of a regular dodecagon is 30∘30^\circ30∘.
Explanation:
Exterior angles of a regular polygon always sum to 360°. This means that if you divide 360° by the number of sides of the polygon, you get the measure of each exterior angle. The key idea here is that a regular polygon has equal angles at every vertex, so all the exterior angles are the same.
This relationship is fundamental to geometry and is used to calculate the measures of angles in polygons such as triangles, squares, pentagons, and more. In the case of the regular dodecagon, which has 12 sides, dividing 360° by 12 gives you 30°, which is the exterior angle at each vertex of the dodecagon.
Each of these exterior angles corresponds to the angle formed by one side of the polygon and the extension of the adjacent side. These angles are helpful in various geometric constructions and proofs, especially when dealing with symmetry and regularity in polygons.
