What polygon is always concave?

Answer 1

While there is no regular polygon that is concave, all types of polygons with 4 sides or more can have irregular concave forms.

Triangles CANNOT be concave. This is because the total of the angle measures in a triangle MUST equal 180 degrees. (The formula for the total angle measures of any polygon is 180*(n-2), where n equals the number of sides in the polygon.) Since any CONCAVE polygon MUST have an angle with a measure greater than 180 degrees, it is impossible for a triangle be concave, since angles CANNOT have negative degree measures.

Also, the mathematical reason that regular polygons cannot be concave relates to a formula similar to the above angle measure formula. Remember that in a regular polygon all of the angles have the same degree measure. So, the formula for the degree measure of any SINGLE angle in a regular polygon is (180*(n-2))/n ( / means to divide). This can be simplified to (180n-360)/n, and it is the same as 180 - 360/n. Since n can't = 0 nor be negative, and since 360 divided by any positive number still = a positive #, 180 MINUS ANY POSITIVE NUMBER WOULD GIVE YOU A NUMBER LESS THAN 180, MEANING THAT NO ANGLE MEASURE IS GREATER THAN 180, WHICH IS REQUIRED IN A CONCAVE POLYGON.