Formulas Involving Polygons Chapter 7 Section 3 By: Alex Pipcho

Polygon Names 8 sides – Octagon 3 sides – Triangle 9 sides- Nonagon 10 sides – Decagon 12 sides – Dodecagon 15 sides –Pentadecagon n sides – n-gon 3 sides – Triangle 4 sides – Quadrilateral 5 sides – Pentagon 6 sides – Hexagon 7 sides – Heptagon

Vocabulary Every segment in the polygon that joins two non-consecutive vertices is a diagonal. Interior angles are formed by two consecutive sides of a polygon. Exterior angles are adjacent and supplementary to an interior angle of the polygon.

Finding Sum of Angles To find the number of degrees in a polygon, draw all the diagonals possible from one vertex. Then count the number of triangles formed and multiply that by 180 (the number of degrees in one triangle). Example: When 2 diagonals are drawn in the figure below, 3 triangles are formed. In conclusion, the sum of the measures of the angles in a pentagon is 3(180) or 540°. But, by using Theorem 55, the sum of the measures of the angles could be found in an easier way.

Theorem 55 The sum Si of the measures of the angles of a polygon with n sides is given by the formula Si = (n-2)180. Example: What is the sum of the measures of the angles in a heptagon? Solution: Use the formula above and substitute 7 for n Si = (7-2)180 = (5)180 = 900°

Theorem 56 If one exterior angel is taken at each vertex, the sum Se of the measures of the exterior angles of a polygon is given by the formula Se = 360. Therefore, the sum of the measures of the exterior angles in any polygon is 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 Example: How many diagonals can be drawn in an 18-gon? Solution: Use the formula above and substitute 18 for n d = 18(18-3) = 270 = 135 diagonals

Regular Polygon Formulas To find the measure of one angle of a regular polygon with n sides, use the following formula: I = (n-2)180 n Example: What is the measure of one angle in a regular nonagon? Solution: Use the formula above and substitute 9 for n I = (9-2)180 9 = 1260 = 140°

Regular Polygon Formulas (Cont.) To find the measure of one exterior angle of a regular polygon with n sides, use the following formula: E = 360 n Example: What is the measure of one exterior angle of a regular octagon? Solution: Use the formula above and substitute 8 for n 8 = 45°

Practice Problems How many sides does a polygon have if the sum of the measures of its angles is 3240°? What is the sum of the measures of the angles of a 31-gon? Given: m A = 85°, m B = 115°, m C = 95°, m D = 100° Find : m E

Practice Problems (Cont.) What is the sum of the measures of the exterior angles, one per vertex, of a decagon? What is the name of a polygon with 65 diagonals? How many diagonals does a 22-gon have? What is the measure of one angle of a regular decagon? What regular polygon has an angle measuring 150°?

Practice Problems (Cont.) 9. What regular polygon has an exterior angle measuring 6°? What is the measure of one exterior angle of a regular octagon? Answers on next slide

Answers to Practice Problems 20 sides 5220° m E = 145° 360° 13-gon 209 144 Dodecagon 60-gon 45°

Works Cited Rhoad, Richard, George Milauskas, and Robert Whipple. Geometry for Enjoyment and Challenge. Evanston, Illinois: McDougal, Littell & Company, 1991. Habeeb, Danielle. “Diagonals in a Polygon.” Geometry for Middle School Teachers Institute. CPTM. 24 May 2008 <http://intermath.coe. uga.edu/tweb/cptm1/dhabeeb/diagonals/ diagonalsinapolygon.htm>.