Regular polygons: angle, number of sides, construction
Find the number of sides of a regular polygon from its angle, test whether it exists, and build a hexagon by chords R and a 2n-gon.
- Palestine: grade 10
- Ukraine: grade 9
- Russia: grade 9
Guide
In Ukrainian Grade 9 geometry (section 6) the angle of a regular n-gon is 180°n − 2n and the central angle is ∠AOB = 360°/n. To find n from an angle, write an equation: the sum of the angles 180°(n − 2) equals 155°n. If n is not a natural number, there is no such polygon.
The side of a regular hexagon equals the radius of its circumscribed circle: a6 = R. So from any point M of the circle we lay off chords equal to the radius one after another. Joining the vertices through one gives a regular triangle. Two perpendicular diameters give a square.
Once a regular n-gon is built, a regular 2n-gon is easy: find the midpoints of all arcs (sides) and draw radii through them. This turns a hexagon into a 12-gon and a square into an 8-gon. Figures are drawn to scale: 1 cell is 1 cm.