Formulas involving Polygons Lesson 7.3

Sums of interior angles

Theorem 55: Sum Si of the measure of the angles of a polygon with n sides is given by the formula Si = (n-2)180

Exterior angles Sum of interior <‘s = 3(180) = 540 2 4 Sum of interior <‘s = 3(180) = 540 Sum of 5 supplementary <‘s = 5(180) = 900 900 - 540 = 360 Total sum of all exterior <‘s = 360 3

Theorem 56 : If one exterior angle is taken at each vertex, the sum Se of the measures of the exterior <‘s of a polygon is given by the formula Se = 360

Theorem 57: The number of diagonals that can be drawn in a polygon of n sides is given by the formula d = n(n-3) 2 Try: draw then do the math!

In what polygon is the sum of the measure of exterior <s, one per vertex, equal to the sum of the measure of the <s of the polygon? Quadrilateral 360 = 360

In what polygon is the sum of the measure of interior <s equal to twice the sum of the measure of the exterior <s, one per vertex? Hexagon: 720 int. = 2(360) ext. 720 = (n-2)(180) 720 = 180n – 360 1080 = 180n n = 6