Unit 7: Polygons. Lesson 7.1: Interior & Exterior Angle Sums of Polygons

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1 Lesson 7.1: Interior & Exterior Angle Sums of Polygons Unit 7: 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. Definition of a Polygon A is plane figure ( -dimensional) that meets the following conditions. 1. It is formed by or segments called. 2. The sides must be lines. 3. Each side intersects other sides, one at each endpoint. 4. The polygon is closed in all the way around with. 5. Each side must end when the next side begins.. Types of Polygons Number of Sides Type of Polygon Number of Sides Type of Polygon Three Octagon Quadrilateral Nine Pentagon Decagon Six Twelve Seven Any other number (n) 1

2 Concave v Convex A polygon is 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 between the two points that the interior of the polygon, then it is. A polygon is 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. polygons have in the sides, or you could say it. 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 Diagonals of a Polygon A of a polygon is a that joins two vertices. A diagonal does not go to the point to it. That would make it a side! Diagonals the polygon to points on the other side. There is typically more than one diagonal. 2

3 Interior Angles of a Polygon The sum of the interior angles of a triangle is. The sum of the interior angles of a quadrilateral is. 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, it should be able to tell you the sum of the interior angles. Pick one and draw all possible from that vertex. Then, count up the number of triangles and multiply by. Theorem 11.1: Polygon Interior Angles Theorem The of the measure of the of a convex n-gon is Example 7.2 Find the sum of the interior angles of the following convex polygons nonagon gon Example 7.3 Find x

4 Exterior Angles An angle is formed by extending each side of a polygon in direction. Make sure they all extend either pointing clockwise or counter-clockwise. Theorem 11.2: Polygon Exterior Angles Theorem The of the measures of the of a convex polygon is. As if you were traveling in a! Example 7.4 Find the sum of the exterior angles of the following convex polygons. 1. Triangle 2. Quadrilateral 3. Pentagon 4. Hexagon 5. Heptagon 6. Dodecagon gon Example 7.5 Find x Lesson 7.1 Homework Lesson 7.1 Interior & Exterior Angle Sums of Polygons Due 4

5 Lesson 7.2: Each Interior & Exterior Angle of a Regular Polygon 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. Regular Polygons A polygon is if all of its are congruent. A polygon is if all of its interior are congruent. A polygon is if it is equilateral and equiangular. Remember: side must be marked with the congruence marks and angle must be marked with the congruence arcs. Example 7.6 Classify the following polygons as equilateral, equiangular, regular, or neither

6 Corollary to Theorem 11.1 The measure of interior angle of a regular n-gon is found using the following: It basically says to take the of the interior angles and by the of to figure out how big each angle is. Example 7.7 Find the measure of each interior angle in the regular polygons. 1. pentagon 2. decagon gon 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 the known by the of of the polygon. That will tell us how many sides it would take to be to the of all the 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: Example 7.8 Determine the number of sides of the regular polygon given one interior angle o o o 6

7 Corollary to Theorem 11.2 Review: What is the sum of the exterior angles of a pentagon? hepatagon? dodecagon? any polygon? Then how would we find the measure of an exterior angle if it were a polygon? Divide by the of exterior angles formed. o Which happens to be the same as the ( ). This can also be worked in reverse to determine the number of sides of a regular polygon the measure of an exterior angle. How? Figure out how that angle measure would go into. Say each exterior angle is. How many exterior angles would it take to get to the total for the exterior angles? So n = Example 7.9 Find the measure of each exterior angle of the regular polygon. 1. octagon 2. dodecagon gon Example 7.10 Determine the number of sides of the regular polygon given the measure of an exterior angle o o o Lesson 7.2 Homework Lesson Each Interior & Exterior Angle of a Regular Polygon Due 7

8 Lesson 7.3: Area and Perimeter of Regular Polygons (Day 1) 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. Parts of a Polygon The of a polygon is the center of the polygon s circle. A is one in that is drawn to go through the of a polygon. The of a polygon is the radius of its circle. Will go from the to a. Central Angle of a Polygon The of a polygon is the angle formed by drawing lines from the to two vertices. Example 7.11 Find the central angle of the following regular polygons. 1. pentagon 2. heptagon 3. decagon gon Reminder of - Postulate 22: Area of a Square Postulate The area of a square is the square of the length of its side. 8

9 Theorem 11.3: Area of an Equilateral Triangle Area of an equilateral triangle is: Example 7.12 Find the area of the equilateral triangles Interior Triangles of a Hexagon A is unique in that it is the only polygon whose form vertices of that are all. Remember, an equilateral triangle is also regular! Example 7.13 Find the area of the regular hexagons Lesson 7.3a Homework Lesson 7.3 Area & Perimeter of Regular Polygons (Day 1) Due 9

10 Lesson 7.3: Area and Perimeter of Regular Polygons (Day 2) (Using Special Triangles & Trigonometry) Perimeter of a Regular Polygon Recall that the is the of the lengths of all the of a figure. Well what is true about the side lengths of a polygon? They are all. So the quickest and way to find the when all sides are congruent is: Do the Equilateral Triangles Still Exist? What is true about triangles? All sides are, and All angles are. o And each angle must be. What is the central angle of a regular pentagon? o Would that central angle help to form an equilateral triangle? What is the central angle of a regular heptagon? o Would that central angle help to form an equilateral triangle? So the equilateral triangles are only formed in. Therefore, there must be another way to find the area of other regular polygons. Apothem The is the of a line segment in a regular polygon drawn: 1. From the of the polygon to one of its. 2. Such that it is to the side. 3. And it the of the polygon. 10

11 Theorem 11.4: Area of a Regular Polygon The area of a regular polygon is found using: Example 7.14 Find the perimeter and area of the regular polygons Lesson 7.3b Homework Lesson 7.3 Area & Perimeter of Regular Polygons (Day 2) Due 11

12 Lesson 7.3: Area and Perimeter of Regular Polygons (Day 3) (Using Special Triangles & Trigonometry - Again) The Apothem and the Central Angle Remember it is necessary to know the length of the when finding the of a regular polygon. So what would happen if the length of the was? Hint: Draw the and what do you see? Because the is a to the side of known length It the in, and It the in. Finding the Area with Only a Known Side Length To find the area of a regular polygon with only a, you must also know the length of the apothem. To do so, create a small using: 1. The. 2. of the. 3. of the given. And then use to solve for the unknown. 12

13 Example 7.15 Find the area of the regular polygons Finding the Area with Only a Known Apothem To find the area of a regular polygon with only a, you must also know the side length. To do so, create a small using: 1. The. 2. of the central angle. 3. of the given side length. And then use to solve for the side length. SOH CAH TOA 13

14 Example 7.16 Find the area of the regular polygons Lesson 7.3b Homework Lesson 7.3 Area & Perimeter of Regular Polygons (Day 3) Due 14