Question

What is one exterior angle of a regular 12-gon measure?

Answer 1

Hint: For solving this question you should know about the exterior angle of a polygon. We know that a circle has a total angle of ${360}^{\circ }$ and we can make that as a dodecagon of 12-gon. For this we have to divide the whole circle in four quadrants and the four quadrants in three parts again. And the exterior angle will be the angle between two continuous parts in a quadrant.

Complete step-by-step solution:
According to our question we have to calculate the exterior angle of a dodecagon (12-gon). Now if we see the diagram of a decagonal, then it is clear as to what a decagonal is, and what is an exterior angle.

If we see the diagram, then it is clear that the decagon is a shape with 10 arms or sides which will make a design like a circle and the angle between two continuous sides is known as the exterior angle of that. For calculating the exterior angle of the dodecagon, we will calculate the interior angle for this.
Now, we know the formula for calculating exterior angle is, 1 exterior angle = ${\displaystyle \frac{{360}^{\circ }}{n}}$, where $n$ is the number of the sides.
For a dodecagon (12-gon) 1 exterior angle = ${\displaystyle \frac{{360}^{\circ }}{12}}={30}^{\circ }$
If you want the size of 1 exterior angle, subtract this from ${180}^{\circ }$, thus,
1 exterior angle = ${180}^{\circ }-{30}^{\circ }={150}^{\circ }$
So, if we calculate the exterior angles of a dodecagon, then,
Number of angles = ${\displaystyle \frac{{360}^{\circ }}{\text{angle}}}$
So, in a dodecagon, the number of angles are 12, so the angle will be,
${\displaystyle \frac{{360}^{\circ }}{12}}={30}^{\circ }$

Note: For calculating the exterior angle of any polygon you should use the formula or we can also calculate this by general division, but it will be an inaccurate method. So, you have to use the formula. And the formula is the number of angles is the ratio of 360 and the angle.