Question

The smallest exterior angle of a polygon is 40degrees the largest exterior angle is 80 degrees a) what is the smallest possible number of sides b) what is the largest possible number of sides

Answer 1

Answer:

a) The smallest possible number of sides of the polygon is a regular polygon with 5 sides

b) The largest possible number of sides of the polygon is a regular polygon with 9 sides

Step-by-step explanation:

a) The smallest exterior angle of the polygon = 40°

The largest exterior angle of the polygon = 80°

Therefore, we have the interior angles are;

The smallest interior angle of the polygon = 180° - 40° = 140°

The largest interior angle of the polygon = 180° - 80° = 100°

Therefore, we have

The interior angles of a regular polygon are given by the following formula;

$Interior \ angle =\dfrac{(n-2) \times 180 ^{\circ}}{n}$

For an interior angle of 100°, we have;

$100^{\circ} =\dfrac{(n-2) \times 180 ^{\circ}}{n}$

100/180 × n = n - 2

2 = n - 100/180 × n = 4/9·n

2 = 4/9·n

n = 18/4 = 4.5

Therefore, given that a square has an interior angle of 90° we have the smallest possible number of sides polygon = 5 sides

b) For 140°, we have;

$140^{\circ} =\dfrac{(n-2) \times 180 ^{\circ}}{n}$

140/180 × n = n - 2

2 = n - 140/180 × n = 2/9·n

2 = 2/9·n

n = 18/2 = 9

n = 9 sides

Therefore, the largest possible number of sides = 9 Sides.