How many sides do these regular
Polygons have if their interior is 30?

Question

How many sides do these regular 
Polygons have if their interior is 30?

Caroline Aucoin · Accepted Answer

#color(blue)("No such polygon")#

The formula for the sum of the interior angles of a regular polygon is given as:

#180^@n-360^@#

We do not know the number of sides, or the sum of the interior angles.

Lets call the sum of the angles S.

#180^@n-360^@=Scolor(white)(888)[1]#

Then one angle is #S/n#

#S/n=30color(white)(888)[2]#

Solving simultaneously:

From #[2]#

#S=30n#

Substituting in #[1]#

#180n-360=30n#

#150n=360=>n=360/150=12/5=2.4#

This is a fractional value, so it can't be the number of sides of a polygon. This means there is no polygon that has interior angles of #30^@#

We could have ascertained this at the beginning. As the number of sides of a polygon increase, the interior angles get larger.

A equilateral triangle is a regular polygon and has interior angles of #60^@#. A square has interior angles of #90^@#. You can see from this that to get an interior angle less than #60^@# you would need to have less than three sides. This is impossible.

Aurora Ayala · Answer

undefined If the interior angle of a regular polygon is 30 degrees, you can use the formula for the interior angle of a regular polygon, which is given by $ \text{Interior angle} = \frac{(n-2) \times 180}{n} $, where $ n $ is the number of sides of the polygon.

Given that the interior angle is 30 degrees, we can set up the equation:

$ 30 = \frac{(n-2) \times 180}{n} $

Solving this equation for $ n $:

$ 30n = (n-2) \times 180 $

$ 30n = 180n - 360 $

$ 360 = 180n - 30n $

$ 360 = 150n $

$ n = \frac{360}{150} $

$ n = 12/5 $

However, since $ n $ represents the number of sides of a polygon, it must be a whole number. Hence, we need to find a common denominator for $ 12/5 $ which is 60.

$ n = \frac{12}{5} \times \frac{12}{12} = \frac{144}{60} $

$ n = 2.4 $

But, since the number of sides must be a whole number, we round $ 2.4 $ to the nearest whole number which is $ 2 $.

Therefore, if the interior angle of a regular polygon is 30 degrees, the polygon would have 2 sides.

How many sides do these regular Polygons have if their interior is 30?