Question

(a) What is the minimum interior angle possible for a regular polygon? Why?
(b) What is the maximum exterior angle possible for regular polygons?

Answer 1

Hint: Use the relation that the interior angle increases when the number of sides increases for any regular polygon to determine the minimum interior angle possible. Similarly, the maximum exterior angle can be calculated using the relation that the measure of exterior angle decreases with the increase in the number of sides for any regular polygon.

Complete step by step Answer:

For any regular polygon, we know that the angle formed by the vertices of the polygon is known as the interior angle, and the angle formed by any side, and the line extended from the next side is known as the exterior angle.
A regular have polygon has equal angles and equal sides.
Also, as we increase the number of sides in the polygon, the interior angle will decrease.
For example, the three-sided regular polygon is an equilateral triangle whose measure of each angle is ${60}^{\circ }$ and the four-sided regular polygon which is a square has a measure of each angle as ${90}^{\circ }$

Then, we can say that the minimum angle is possible for the polygon with the least number of sides.
The smallest polygon is a polygon with 3 sides.
Also, we know that the measure of each angle of the regular polygon, that is an equilateral triangle, is of ${60}^{\circ }$
Now, we have to find the maximum exterior angle of a regular polygon.
Now, we know that the measure of exterior angle decreases when the number of sides increases.
For example, the measure of an exterior angle of a regular polygon with three sides is ${120}^{\circ }$ whereas the measure of each exterior angle of a four-sided regular polygon is ${90}^{\circ }$

Hence, the maximum exterior is possible for the polygon with the least number of sides.
Therefore, the exterior angle of the equilateral triangle is the maximum exterior angle of any polygon.

Note: Students must know that the smallest possible polygon is of three sides and no polygon exists with sides less than 3. Also, in a regular polygon, the angles and sides are equal. The measure of each interior angle of the equilateral triangle is ${60}^{\circ }$ and the measure of each exterior angle of an equilateral triangle is ${120}^{\circ }$