EXPLORYNG POLYGONS AND AREA Objectives: In this chapter you will: • Find the measures of interior and exterior angles of polygons • Find areas of polygons and circles • Solving problems involving geometric probability • Determine characteristics of networks and • Solve problems from real world 1 GEOMETRY THEN AND NOW • Polygons have always been used in structures. In the 1400s, the first structure that resembled a roller coaster was built in Russia. Its supporting framework was made of triangles and parallelograms, much like today’s coasters. Interlocking polygons make up the structure of modern geodesic domes like Spaceship Earth at Walt Disney World’s EPCOT center. This spherical shell is 18 stories tall. • How are polygons used in the designs of buildings in your community? 2 Polygons Are polygons useful in our world? Do you see any need to understand what polygons are or how the knowledge of polygons can help you as you journey through life? Take a look around your world…can you really imagine a world without polygons. 3 4 Did you know? • Seventeen-year-old (then) Ryan Morgan of Baltimore, Maryland, has a geometry theorem named after him. As a high school freshman, Ryan investigated a theorem called (Marino)Walter’s Theorem. This theorem states that if you divide the sides of a triangle into equal thirds and draw lines from each division point to the opposite vertex, the lines form a hexagon inside the triangle. 5 Did you know? • (Mathematics Teacher 1993, Maushard 1994, Morgan 1994) • Furthermore, the area of the hexagon is one tenth of the triangle’s area. Ryan began experimenting with dividing the sides of triangles by other numbers. Ryan eventually found that dividing a triangle’s sides by any odd number and connecting the endpoints of each middle segment to the opposite vertex also forms a hexagon. 6 Did you know? • The area of the hexagon is always proportional to the area of the original triangle. Once his conjecture was proved by experts, it was officially named “ “Morgan’s Theorem” in his honor. 7 8 11.1 Angle Measures in Polygons NCSCOS: 2.02; 2.03 9 Essential Question: • How can we use measures of angles of polygons to solve real-life problems? • How do we find the measures of interior and exterior angles of polygons? 10 Special types of Polygons • Convex- no line that contains a side of a polygon goes through its interior • Concave- opposite of a convex • Equilateral- all sides are  • Equiangular- all angles are  • Regular polygon- equilateral and equiangular Measures of Interior and Exterior Angles • In lesson 6.1, you found the sum of the measures of the interior angles of a quadrilateral by dividing the quadrilateral into two triangles. You can use this triangle method to find the sum of the measures of the interior angles of any convex polygon with n sides, called an n-gon. • (Okay – n-gon means any number of sides – including 11—any given number (n). 12 If all the s in a Δ add up to 180o and all the s in a quadrilateral add o up to 360 , what about a pentagon? How about a hexagon? 2 * 180 = 3600 3 * 180 = 540o 4 * 180 = 720o 13 Measures of Interior and Exterior Angles • For instance . . . Complete this table Polygon Triangle # of sides 3 Quadrilateral # of triangles 1 Sum of measures of interior ’s 1●180=180 2●180=360 Pentagon Hexagon Nonagon (9) n-gon n 14 Polygon Interior Angles Theorem • The sum of the measures of the interior angles of a convex n-gon is (n – 2) ● 180 • COROLLARY: The measure of each interior angle of a regular n-gon is: 1 n or ● (n-2) ● 180 ( n  2)(180) n 15 Ex: What is the sum of the measures of the interior s of a dodecagon? First, how many sides does a dodecagon have? n = 12 180(12-2) = 180(10) = 1800o 16 Ex: Find the value of x. 114o 105o Sum of all s is 540o 114+105+102+135=456o 540 – 456 = 84o So, x = 84 102o 17 Ex: The measure of each interior of a regular polygon is 165o. How many sides does the polygon have? 180(n  2)  165 n 180(n-2) = 165n 180n-360 = 165n -360 = -15n 24 = n 24 sides 18 Notes • The diagrams on the next slide show that the sum of the measures of the exterior angles of any convex polygon is 360. You can also find the measure of each exterior angle of a REGULAR polygon. 19 Copy the item below. 20 Thm 11.2 – Polygon Exterior s thm The sum of the measures of the exterior s of a convex polygon, one at each vertex, is 360o. 2 m1 + m2 + m3 + m4 + m5 = 360o 1 3 5 4 21 Ex: Solve for y. y 2y 2y y 2y + y + 2y + y = 360 6y = 360 y = 60o 22 Corollary to thm 11.2 • The measure of each exterior  of a regular n-gon is 360 n 23 Ex: Solve for x. xo n=6 360/n = x 360/6 = x 60o = x This is a regular hexagon. 24 Using Angle Measures in Real Life Ex.: Finding Angle measures of a polygon 25 Using Angle Measures in Real Life Ex.: Using Angle Measures of a Regular Polygon 26 Using Angle Measures in Real Life Ex.: Using Angle Measures of a Regular Polygon 27 Using Angle Measures in Real Life Ex. : Using Angle Measures of a Regular Polygon Sports Equipment: If you were designing the home plate marker for some new type of ball game, would it be possible to make a home plate marker that is a regular polygon with each interior angle having a measure of: a. 135°? b. 145°? 28 Using Angle Measures in Real Life Ex. : Finding Angle measures of a polygon 29 Summarizer • Explain in words how to find the measure of each interior angle and each exterior angle in a regular polygon.