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Unit 7

Unit 7. Polygons. Lesson 7.1. Interior & Exterior Angle Sums of Polygons. Lesson 7.1 Objectives. Calculate the sum of the interior angles of a polygon. (G1.5.2) Calculate the sum of the exterior angles of a polygon. (G1.5.2) Classify different types of polygons.

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Unit 7

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  1. Unit 7 Polygons

  2. Lesson 7.1 Interior & Exterior Angle Sums of Polygons

  3. Lesson 7.1 Objectives • Calculate the sum of the interior angles of a polygon. (G1.5.2) • Calculate the sum of the exterior angles of a polygon. (G1.5.2) • Classify different types of polygons.

  4. Definition of a Polygon • A polygon is plane figure (two-dimensional) that meets the following conditions. • It is formed by three or more segments called sides. • The sides must be straight lines. • Each sideintersectsexactly twoother sides, one at each endpoint. • The polygon is closed in all the way around with no gaps. • Each side must end when the next side begins. No tails. Polygons Not Polygons Too Many Intersections No Curves No Gaps No Tails

  5. Types of Polygons Triangle Quadrilateral Pentagon Hexagon Heptagon Octagon Nonagon Decagon Dodecagon n-gon

  6. A polygon is convex if no line that contains a side of the polygon contains a point in the interior of the polygon. Take any two points in the interior of the polygon. If you can draw a line between the two points that neverleave the interior of the polygon, then it is convex. A polygon is concave if a line that contains a side of the polygon contains a point in the interior of the polygon. Take any two points in the interior of the polygon. If you can draw a line between the two points that does leave the interior of the polygon, then it is concave. Concave polygons have dentsin the sides, or you could say it caves in. Concave v Convex

  7. Example 7.1 Determine if the following are polygons or not.If it is a polygon, classify it as concave or convex and name it based on the number of sides. Yes No! Concave Octagon Yes No! Concave Hexagon Concave Yes Heptagon Yes Convex Pentagon

  8. Diagonals of a Polygon • A diagonalof a polygon is a segment that joins two nonconsecutive vertices. • A diagonal does not go to the point next to it. • That would make it a side! • Diagonals cut across the polygon to all points on the other side. • There is typically more than one diagonal.

  9. Interior Angles of a Polygon • The sum of the interior angles of a triangle is • 180o • The sum of the interior angles of a quadrilateral is • 360o • The sum of the interior angles of a pentagon is • ??? • The sum of the interior angles of a hexagon is • ??? • By splitting the interior into triangles, it should be able to tell you the sum of the interior angles. • Pick one vertex and draw all possible diagonals from that vertex. • Then, count up the number of triangles and multiply by 180o. 180o 360o 540o 720o

  10. Theorem 11.1:Polygon Interior Angles Theorem • The sum of the measure of the interior angles of a convexn-gon is n = number of sides

  11. Example 7.2 Find the sum of the interior angles of the following convex polygons. • nonagon • 17-gon

  12. Example 7.3 Find x.

  13. Exterior Angles 4 3 5 2 1 • An exterior angle is formed by extending each side of a polygon in one direction. • Make sure they all extend either pointing clockwise or counter-clockwise.

  14. Theorem 11.2:Polygon Exterior Angles Theorem 4 3 5 2 1 • The sum of the measures of the exterior angles of a convexpolygon is 360o. • As if you were traveling in a circle!  1 +  2 +  3 +  4 +  5 = 360o

  15. Example 7.4 Find the sum of the exterior angles of the following convex polygons. • Triangle • 3600 • Quadrilateral • 3600 • Pentagon • 3600 • Hexagon • 3600 • Heptagon • 3600 • Dodecagon • 3600 • 17-gon • 3600

  16. Example 7.5 Find x.

  17. Lesson 7.1 Homework • Lesson 7.1 – Interior & Exterior Angle Sums of Polygons • Due Tomorrow

  18. Lesson 7.2 Each Interior & Exterior Angle of a Regular Polygon

  19. Lesson 7.2 Objectives • Calculate the measure of each interior angle of a regular polygon. (G1.5.2) • Calculate the measure of each interior angle of a regular polygon. (G1.5.2) • Determine the number of sides of a regular polygon based on the measure of one interior angle. • Determine the number of sides of a regular polygon based on the measure of one exterior angle.

  20. Regular Polygons • A polygon is equilateral if all of its sides are congruent. • A polygon is equiangular if all of its interior angles are congruent. • A polygon is regular if it is both equilateral and equiangular. Remember: EVERY side must be marked with the same congruence marks andEVERY angle must be marked with the same congruence arcs.

  21. Example 7.6 Classify the following polygons as equilateral, equiangular, regular, or neither.

  22. Corollary to Theorem 11.1 • The measure of each interior angle of a regularn-gon is found using the following: Sum of the Interior Angles Divided equally into nangles. It basically says to take the sum of the interior angles and divide by the number of sides to figure out how big each angle is.

  23. Example 7.7 Find the measure of each interior angle in the regular polygons. • pentagon decagon 17-gon

  24. Finding the Number of Sides • By knowing the measurement of one interior angle of a regular polygon, we can determine the number of sides of the polygon as well. • How? • Since we know that all angles are going to have the same measure we will multiply the known angle by the number of sides of the polygon. • That will tell us how many sides it would take to be set equal to the sum of all the interior angles of the polygon. • However, since we do not know the number of sides of the polygon, nor do we know the total sum of the interior angles of that polygon we are left with the following formula to work with:

  25. Example 7.8 Determine the number of sides of the regular polygon given one interior angle. • 120o • 140o • 147.27o

  26. Review: What is the sum of the exterior angles of a pentagon? 3600 hepatagon? 3600 dodecagon? 3600 any polygon? 3600 Then how would we find the measure of an exterior angle if it were a regular polygon? Divide 360o by the number of exterior angles formed. Which happens to be the same as the number of sides (n). This can also be worked in “reverse” to determine the number of sides of a regular polygon given the measure of an exterior angle. How? Figure out how many times that angle measure would go into 360o. Say each exterior angle is 1200. How many exterior angles would it take to get to the total for the exterior angles? 360120 3 So n = 3  Corollary to Theorem 11.2

  27. Example 7.9 Find the measure of each exterior angle of the regular polygon. • octagon dodecagon 15-gon

  28. Example 7.10 Determine the number of sides of the regular polygon given the measure of an exterior angle. • 72o • 36o • 27.69o

  29. Lesson 7.2 Homework • Lesson 7.2 - Each Interior & Exterior Angle of a Regular Polygon • Due Tomorrow

  30. Lesson 7.3 Day 1: Area and Perimeter of Regular Polygons

  31. Lesson 7.3 Objectives • Calculate the measure of the central angle of a regular polygon. • Identify an apothem • Calculate the perimeter and area of a regular polygon. (G1.5.1) • Utilize trigonometry to find missing measurements in a regular polygon.

  32. Parts of a Polygon • The centerof a polygon is the center of the polygon’s circumscribed circle. • A circumscribed circle is one in that is drawn to go through all the vertices of a polygon. • The radius of a polygon is the radius of its circumscribed circle. • Will go from the center to a vertex. r

  33. Central Angle of a Polygon • The central angle of a polygon is the angle formed by drawing lines from the center to two consecutive vertices. • This is found in a regular polygon by: • That is because the total degrees traveled around the center would be like a circle. • Then divide that by the number of sides because that determines how many central angles could be formed.

  34. Example 7.11 Find the central angle of the following regular polygons. • pentagon • heptagon • decagon • 18-gon

  35. Reminder of -Postulate 22:Area of a Square Postulate • The area of a square is the square of the length of its side. s

  36. Theorem 11.3:Area of an Equilateral Triangle • Area of an equilateral triangle is: s

  37. Example 7.12 Find the area of the equilateral triangles.

  38. Interior Triangles of a Hexagon 5 4 3 6 2 • A regularhexagon is unique in that it is the only polygon whose central angles form vertices of interior triangles that are all equilateral. • Remember, an equilateral triangle is alsoregular! Remember that every side of the interior triangles form a radius of the circle around the vertices. AND ALL RADII ARE CONGRUENT! 1 • So to find the area of a regular hexagon would be like finding the area of SIXequilateral triangles!

  39. Example 7.13 Find the area of the regular hexagons.

  40. Lesson 7.3a Homework Lesson 7.3 – Area & Perimeter of Regular Polygons (Day 1) Due Tomorrow

  41. Lesson 7.3 Day 2: Area and Perimeter of Regular Polygons (Using Special Triangles & Trigonometry)

  42. Perimeter of a Regular Polygon Recall that the perimeter is the sum of the lengths of all the sides of a figure. Well what is true about the side lengths of a regular polygon? They are all equilateral. So the quickest and best way to find the perimeter when all sides are congruent is: s = side length n = number of sides

  43. Do the Equilateral Triangles Still Exist? • What is true about equilateral triangles? • All sides are congruent, and… • All angles are congruent. • And each angle must be 60o • What is the central angle of a regular pentagon? • 72o • Would that central angle help to form an equilateral triangle? • No, because all angles must be 60o. • What is the central angle of a regular heptagon? • 51.43o • Would that central angle help to form an equilateral triangle? • No, because all angles must be 60o. • So the equilateraltriangles are only formed in hexagons. • Therefore, there must be another way to find the area of other regular polygons.

  44. Apothem • The apothem is the length of a line segment in a regular polygon drawn: • From the center of the polygon to one of its sides. • Such that it is perpendicular to the side. • And it bisects the side of the polygon. a

  45. Theorem 11.4: Area of a Regular Polygon • The area of a regular polygon is found using: P = perimeter P = n•s And how do wefind theperimeter? a = apothem s = side length ( ) n = number of sides

  46. Example 7.14 Find the perimeter and area of the regular polygons.

  47. Lesson 7.3b Homework Lesson 7.3 – Area & Perimeter of Regular Polygons (Day 2) Due Tomorrow

  48. Lesson 7.3 Day 3: Area and Perimeter of Regular Polygons (Using Special Triangles & Trigonometry - Again)

  49. The Apothem and the Central Angle Remember it is necessary to know the length of the apothem when finding the area of a regular polygon. A = 1/2a•n•s So what would happen if the length of the apothem was unknown? Hint: Draw the central angle and what do you see? Because the apothem is a perpendicular bisector to the side of known length It divides the side in half, and It divides the central angle in half. a

  50. Finding the Area with Only a Known Side Length To find the area of a regular polygon with only a known side length, you must also know the length of the apothem. To do so, create a small right triangle using: The apothem. Half of the central angle. Half of the given side length. And then use trigonometry to solve for the unknown apothem. SOH CAH TOA a 6 Half the given side length

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