Unit 7 Polygons

Interior & Exterior Angle Sums of 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.

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 side intersects exactly two other 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

Types of Polygons Triangle Quadrilateral Pentagon Hexagon Heptagon Number of Sides Type of Polygon 3 4 5 6 7 8 9 10 12 n Triangle Quadrilateral Pentagon Hexagon Heptagon Octagon Nonagon Decagon Dodecagon n-gon

Concave v Convex 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 never leave 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 dents in the sides, or you could say it caves in.

Example 7.1 Yes No! Concave Octagon Yes No! Concave Hexagon Concave 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. No! Yes Concave Octagon Yes No! Concave Hexagon Concave Yes Heptagon Yes Convex Pentagon

Diagonals of a Polygon A diagonal of 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.

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

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

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

Example 7.3 Find x.

Exterior Angles 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. 1 2 3 4 5

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

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

Example 7.5 Find x.

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

Each Interior & Exterior Angle of a Regular Polygon 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) 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 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 and EVERY angle must be marked with the same congruence arcs.

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

Corollary to Theorem 11.1 The measure of each interior angle of a regular n-gon is found using the following: Sum of the Interior Angles Divided equally into n angles. 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.

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

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

Corollary to Theorem 11.2 Review: What is the sum of the exterior angles of a pentagon? 3600 hepatagon? dodecagon? any polygon? 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 

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

Example 7.10 Determine the number of sides of the regular polygon given the measure of an exterior angle. 72o 45o 27.69o

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

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

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.

Theorem 11.3: Area of an Equilateral Triangle Area of an equilateral triangle is A = ¼ (√3) s2 Take ¼ times the length of a side squared and write in front of √3. Be sure to simplify if possible. s

Parts of a Polygon The center of 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

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 by 360/n 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. 600

Apothem The apothem is the distance from the center to any side of the polygon. Not to the vertex, but to the center of the side. The height of a triangle formed between the center and two consecutive vertices of the polygon. a

Theorem 11.4: Area of a Regular Polygon The area of a regular n-gon with side length s is half the product of the apothem and the perimeter. A = 1/2aP A stands for area a stands for apothem P stands for perimeter of the n-gon Found by finding the side length and multiplying by the number of sides A = 1/2a(ns) n stands for the number of sides s stands for the length of one side

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

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