three regular polygons meet at point A. How many sides does the largest polygon have?

Question

the smaller ones are a triangle and a nonagon

Answer 1

To find the internal angle t of a regular n-sided polygon, divide 360 by the number of sides, then subtract from 180, i.e.:

t = 180 - ( 360 / n )

Thus for a triangle, the internal angle is: t=180-(360/3)=180-120=60d. For a nonagon, t=180-(360/9)=180-40=140d. Assuming that all three regular polygons occupy all 360 degrees around the point (it wasn't stated, but then the problem would not be solveable), then the remaining angle would be 360-60-140=160d.

We can then simply invert the same equation to find the number of sides for a regular n-sided polygon with an internal angle of t, i.e.:

n = 360 / ( 180 - t )

So a polygon with an internal angle of 160d would have n=360/(180-160)=360/20=18 sides.

Answer 2

All regular polygons have interior angles that measure [180(n-2)]/n

A regular (equilateral) triangle has interior angle = 60 degrees
A regular nonagon has interior angle = [180(9-2)]/9 = 180*7/9 = 140 degrees

That leaves 360 - (60 + 140) = 160 degrees for the interior angle of the third polygon.

160 = [180(n-2)]/n
160n = 180n - 360
-20n = -360
n = 18

The third polygon is an eighteen-sided octadecagon

Answer 3

Regular Triangle - 60 deg.
Regular 9-gon - 140 deg.
That leaves 160 deg. for the last one.
an 18-gon (octodecagon?) has 160 deg. per side.