⁦Regular polygons and the excircle of a triangle⁩

⁦In Ukrainian Grade 9 geometry⁩ (§⁦6) a regular polygon can be inscribed in a circle and circumscribed about a circle, and these circles share a centre⁩. ⁦Let R be the radius of the circumscribed circle, r the radius of the inscribed circle (the apothem) and n the number of sides⁩.

⁦Then a_n = ٢Rsin(١٨٠°/n), r = Rcos(١٨٠°/n), and in terms of r, a_n = ٢rtan(١٨٠°/n)⁩. ⁦The area is S = ١٢Pr, where P is the perimeter⁩.

⁦The excircle of a triangle touches one side and the extensions of the other two⁩. ⁦Its radius is r_a = Sp − a, where S is the area, p the half-perimeter and a the side the circle touches⁩.

⁦The figures are drawn to scale⁩: ⁦1 square is 1 cm⁩.