Notes for Angles of Polygons | Study Guide

University:
Massachusetts Institute of Technology
Course:
18.089 | Review of Mathematics
Academic year:

2025
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0

Pages:

113
Author:

Lakshita Dokku

Unit 7: Polygons & Circles Unit 7: Polygons Poly-gons Poly: many gons: angles Polygon A plane closed figure made up of line segments # of Sides 3 4 5 6 7 8 9 10 12 Polygon Triangle Quadrilateral Pentagon Hexagon Heptagon Octagon Nonagon Decagon Dodecagon > Hendecagon: 11 sides > Triskaidecagon: 13 sides > Tetrakaidecagon: 14 sides > Pentadecagon: 15 sides > Hexakaidecagon: 16 sides > Heptadecagon: 17 sides > Octakaidecagon: 18 sides > Enneadecagon: 19 sides > Icosagon: 20 sides > Hectagon: 100 sides > Chiliagon: 1000 sides > Myriagon: 10,000 sides > Megagon: 1,000,000 sides > Hendecagon: 11 sides > Hectagon: 100 sides > Triskaidecagon: 13 sides > Chiliagon: 1000 sides > Tetrakaidecagon: 14 sides > Myriagon: 10,000 sides > Pentadecagon: 15 sides > Megagon: 1,000,000 sides > Hexakaidecagon: 16 sides > Heptadecagon: 17 sides > Octakaidecagon: 18 sides > Enneadecagon: 19 sides > Icosagon: 20 sides Remembering allRE the names???m > 11-gon: 11 sides > 100-gon: 100 sides > 13-gon: 13 sides > 1000-gon: 1000 sides > 14-gon: 14 sides > 10,000-gon: 10,000 sides > 15-gon: 15 sides > 1,000,000-gon: 1,000,000 sides > 16-gon: 16 sides > 17-gon: 17 sides > 18-gon: 18 sides > 19-gon: 19 sides > 20-gon: 20 sides n-gon 54-gon A polygon with 54 sides Polygons Regular Polygons Irregular Polygons Polygons Regular Polygons Irregular Polygons Regular Polygons Polygons that have all sides congruent and all angles congruent. A regular triangle is called an ________________________ A regular triangle is called an equliateral triangle A regular quadrilateral is called a __________________ A regular quadrilateral is called a square . A regular pentagon is called a __________________ A regular pentagon is called a regular pentagon . Irregular Polygon A polygon with angles that are not all congruent or sides that are not all congruent. Polygon Activity For each polygon, with a ruler, draw as many non-overlapping diagonals as you can. The diagonals cannot leave the interior of the polygon. Polygon Activity Diagonals are segments that connect non-adjacent vertices. Polygon Activity NO GOOD!!! They overlap. Polygon Activity THIS IS GOOD!!! They do not overlap. Polygon Activity NO GOOD!!!! Leaves the interior of the polygon. Polygon Activity For each polygon, with a ruler, draw as many non-overlapping diagonals as you can. The diagonals cannot leave the interior of the polygon. Polygon Activity How many triangles will a 58 sided polygon have? 1 1 2 1 2 3 1 2 3 4 2 3 1 4 5 2 3 1 4 5 6 1 1 2 2 1 3 1 2 3 4 2 1 3 4 5 2 1 3 4 5 6 If you know the number of sides for a given polygon how can you determine the number of non-overlapping triangles that exist in the polygon? If you know the number of sides for a given polygon how can you determine the number of non-overlapping triangles that exist in the polygon? n-2 *where n is the # of sides 1 1 2 1 1 180° 2 1 1 180° 2 180(2) = 360° 1 2 3 1 2 3 4 1 2 3 180(3) = 540° 1 2 3 4 1 2 3 180(3) = 540° 1 2 3 4 180(4) = 720° 2 3 1 4 5 2 3 1 4 5 6 2 3 1 4 5 180(5) = 900° 2 3 1 4 5 6 2 3 1 4 5 180(5) = 900° 2 3 1 4 5 6 180(6) = 1080° # of Sides Non-Overlapping Triangles Angle Sum 3 3-2=1 180(1) = 180° 4 4-2=2 180(2) = 360° 5 5-2=3 180(3) = 540° 6 6-2=4 180(4) = 720° 7 7-2=5 180(5) = 900° 8 8-2=6 180(6) = 1080° # of Sides Non-Overlapping Triangles Angle Sum 3 3-2=1 180(1) = 180° 4 4-2=2 180(2) = 360° 5 5-2=3 180(3) = 540° 6 6-2=4 180(4) = 720° 7 7-2=5 180(5) = 900° 8 8-2=6 180(6) = 1080° n # of Sides Non-Overlapping Triangles Angle Sum 3 3-2=1 180(1) = 180° 4 4-2=2 180(2) = 360° 5 5-2=3 180(3) = 540° 6 6-2=4 180(4) = 720° 7 7-2=5 180(5) = 900° 8 8-2=6 180(6) = 1080° n n-2 # of Sides Non-Overlapping Triangles Angle Sum 3 3-2=1 180(1) = 180° 4 4-2=2 180(2) = 360° 5 5-2=3 180(3) = 540° 6 6-2=4 180(4) = 720° 7 7-2=5 180(5) = 900° 8 8-2=6 180(6) = 1080° n n-2 180(n - 2) # of Sides Non-Overlapping Triangles Angle Sum 3 3-2=1 180(1) = 180° 4 4-2=2 180(2) = 360° 5 5-2=3 180(3) = 540° 6 6-2=4 180(4) = 720° 7 7-2=5 180(5) = 900° 8 8-2=6 180(6) = 1080° n n-2 180(n - 2) Angle Sum of a Polygon 180(n - 2) Classifying Polygons Regular v. Irregular Regular Octagon Irregular Octagon Convex Polygon A polygon is convex when no line that contains a side of the polygon contains a point in the interior of the polygon. Convex Polygon A polygon is convex when no line that contains a side of the polygon contains a point in the interior of the polygon. Convex Polygon A polygon is convex when no line that contains a side of the polygon contains a point in the interior of the polygon. Convex Polygon A polygon is convex when no line that contains a side of the polygon contains a point in the interior of the polygon. Convex Polygon A polygon is convex when no line that contains a side of the polygon contains a point in the interior of the polygon. Convex Polygon A polygon is convex when no line that contains a side of the polygon contains a point in the interior of the polygon. Convex Polygon A polygon is convex when no line that contains a side of the polygon contains a point in the interior of the polygon. Convex Polygon A polygon is convex when no line that contains a side of the polygon contains a point in the interior of the polygon. Convex Polygon A polygon is convex when no line that contains a side of the polygon contains a point in the interior of the polygon. Concave Polygon A polygon is concave when at least one line that contains a side of the polygon contains a point in the interior of the polygon. Concave Polygon A polygon is concave when at least one line that contains a side of the polygon contains a point in the interior of the polygon. Concave Polygon A polygon is concave when at least one line that contains a side of the polygon contains a point in the interior of the polygon. Concave Polygon At least one diagonal of a concave polygon will leave the interior of the polygon. Angle Sum of a Polygon 180(n - 2) *Works for all polygons, convex and concave Find the angle sum of the polygon below. Angle Sum of a Polygon 180(n - 2) n = # of sides Find the angle sum of the polygon below. Angle Sum of a Polygon 180(n - 2) n = # of sides If the angle sum of the regular octagon is 1080°, then find the measure of . If the angle sum of the regular octagon is 1080°, then find the measure of . If the angle sum of the regular octagon is 1080°, then find the measure of . Measure of One Interior Angle of a Regular Polygon Measure of One Interior Angle of a REGULAR Polygon Exterior Angle If the measure of one interior angle of the regular octagon is 135°, then find the measure of . If the measure of one interior angle of the regular octagon is 135°, then find the measure of . What is the sum of all of the exterior angles of a regular octagon? 45° 45° 45° 45° 45° 45° ° 45° 45° What is the sum of all of the exterior angles of a regular octagon? 45° 45° 45° 45° 45° 45° ° 45° 45° Exterior Angle Sum Exterior Angle Sum Exterior Angle Sum 3(120°) = 360° Exterior Angle Sum Exterior Angle Sum Exterior Angle Sum 4(90°) = 360° Exterior Angle Sum Polygon Int. Angle Sum Int. Ang. Ext. Ang Ext. Ang. Sum Reg Triangle 180° 60° 120° 360° Reg Quad 360° 90° 90° 360° Reg Octagon 1080° 135° 45° 360° Exterior Angle Sum Exterior Angle Sum Exterior Angle Sum Exterior Angle Sum = 180n - 180(n - 2) Exterior Angle Sum Exterior Angle Sum = 180°n - 180°(n - 2) Exterior Angle Sum = 180°n - 180°n + 360° Exterior Angle Sum Exterior Angle Sum = 180°n - 180°(n - 2) Exterior Angle Sum = 180°n - 180°n + 360° Exterior Angle Sum = 360° Exterior Angle Sum We were using regular polygons. What about irregular polygons? If the sum of the exterior angles of any polygon is always 360°, then we can use this fact to find the measure of one exterior angle measure of any regular polygon. Measure of One Exterior Angle of a Regular Polygon Measure of One Exterior Angle of a REGULAR Polygon Find the measure of . Find the measure of . Central Angle Measure Angles of Polygons Interior Angles Sum Exterior Angle Measure AND 180(n - 2) Central Angle of a Regular Polygon n = # of sides 360° Interior Angle Measure n of a Regular Polygon Exterior Angles Sum 180(n - 2) n 360° n = # of sides Homework: Complete Practice 7.1 Problems #1, 2, 7, 8, 10, 14, 18, 20, 21, 22, 23, 27, 33, 35