One exterior angle of a regular polygon measures 40. What is the sum of the polygon's interior angle measures? Please show working.

Question

Violet Aldrich · Accepted Answer

#1260^o# if one exterior angle #=40^0# this gives us the following information number of sides #=360/40=9" "#sides one interior angles #=180-40=140^0# so the sum of interior angles#=9xx140# #=1260^0#

Noah Atkinson · Answer

undefined The sum of the interior angles of a polygon can be calculated using the formula:

Sum of interior angles = (n - 2) * 180 degrees

Where 'n' represents the number of sides in the polygon.

To find the sum of the interior angles of a regular polygon when one exterior angle measures 40 degrees, we can use the fact that the sum of the interior and exterior angles at each vertex of a polygon is 180 degrees.

Given that one exterior angle measures 40 degrees, the corresponding interior angle measures 180 - 40 = 140 degrees.

Since the polygon is regular, all interior angles are congruent. Thus, each interior angle measures 140 degrees.

Now, let's find the number of sides (n) using the relationship between the exterior angle and the number of sides in a polygon:

Exterior angle = 360 / n

Substituting the given exterior angle:

40 = 360 / n

Solving for n:

n = 360 / 40
n = 9

Now, plug the value of n into the formula for the sum of interior angles:

Sum of interior angles = (n - 2) * 180
Sum of interior angles = (9 - 2) * 180
Sum of interior angles = 7 * 180
Sum of interior angles = 1260 degrees

Therefore, the sum of the interior angle measures of the polygon is 1260 degrees.