How many sides does an equiangular polygon have if each angle is 174?

Question

How many sides does an equiangular polygon have if each angle is 174?

Answer 1

60

Answer 2

Let TS = total number of sides in the polygon

Let TD = total number of degrees in the polygon

Let A = the angle inside the polygon

For any polygon (i.e. more than two sides), the number of degrees in the polygon is defined as

TD = 180 * (TS - 2)

(a triangle has 3 sides and 180 total degrees, a rectangle has four sides and 360 total degrees, and so forth.)

For any regular polygon, we know that the internal angle is equal to the total degrees divided by the number of sides:

A = TD/TS

multiplying both sides by TS, we get

A * TS = TD

but from the first equation, we know that

TD = 180 * (TS - 2)

so set them equal to one another:

A * TS = 180 * (TS - 2)

We know that A = 174, so substituting and multiplying, we get

174 * TS = (180 * TS) - 360

or

360 = 6 * TS

or

360/6 = TS

or

60 = TS

So for a regular polygon with internal angles of 174, we know it has 60 sides. QED.

Answer 3

360°/(180° - 174°) = 60 sides