Formulas involving PolygonsLesson 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

Sum of 5 supplementary <‘s = 5(180)

= 900

900 - 540 = 360

Total sum of all exterior <‘s = 360

1

2

3

45

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