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.
- فلسطين: الصف ١٠
- أوكرانيا: الصف ٩
- روسيا: الصف ٩
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الدليل
In Ukrainian Grade 9 geometry (section 6) the angle of a regular n-gon is ١٨٠°n − ٢n and the central angle is ∠AOB = ٣٦٠°/n. To find n from an angle, write an equation: the sum of the angles ١٨٠°(n − ٢) equals ١٥٥°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: a٦ = 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.