Factors of 360

You saw that all the angles of symmetry of a figure are multiples of the smallest one. And 360° is always an angle of symmetry. Put these two facts together: 360 must be a multiple of the smallest angle. So the book asks, true or false: if the smallest angle of symmetry is a whole number of degrees, it is a factor of 360. It is true. (Chapter 5's factors come back here!)

Can a figure have 45° as its smallest angle of symmetry? 360 ÷ 45 = 8, a whole number, so 45 is a factor of 360. Yes: the angles would be 45°, 90°, 135°, 180°, 225°, 270°, 315° and 360°, which is 8 angles of symmetry. What about 17°? 17 × 21 = 357 and 17 × 22 = 374, so no multiple of 17 lands exactly on 360. 17 is not a factor of 360, so 17° cannot be the smallest angle of symmetry.

Another puzzle from the book: in a figure, 60° is an angle of symmetry, and the figure has two angles of symmetry less than 60°. What is the smallest? The angles are multiples of the smallest one, and 60° must be one of those multiples. Two multiples come before 60°, so 60° is the third: smallest, twice the smallest, then three times the smallest = 60°. So the smallest angle is 60° ÷ 3 = 20°, and the angles below 60° are 20° and 40°.