in a regular polygon the ratio of the measure of the interior angle to its adjecent exterior angle is 29 to 1. how many sides does the polygon have?

Question

in a regular polygon the ratio of the measure of the interior angle to its adjacent exterior angle is 29 to 1. how many sides does the polygon have?

Answer 1

29:1    

29x+x=180     (straight angle)

30x=180

x=180/30

x=6

interior angle=29x=29(6)=174 degrees

interior angle=180(n-2)/n      (where n is the number of sides)

174=180(n-2)/n

174n=180n-360

180n-174n=360

6n=360

n=360/6

n=60

The polygon has 60 sides.

Answer 2

A regular polygon with n sides can be split into n isosceles triangles by joining the centre of the polygon to the endpoints of the sides. These joins are radii of the circumscribed circle. The angle at the centre for each of these measures 360/n degrees.

If the other two angles have a measure θ, then 2θ+360/n=180, 2θ=180-360/n. 2θ is also the size of the interior angle so its supplement, the exterior angle, is 180-2θ=360/n.

Therefore 180-360/n=29(360/n), 30(360/n)=180, 360/n=6, so n=60. The polygon has 60 sides.

CHECK

The angle at the centre is 6° so the interior angle is 180-6=174° and the exterior angle is 6°. 174/6=29.