Polygons Use a ruler to draw 3 different POLYGONS Discuss with a partner what a POLYGON is Write down the key qualities a POLYGON has Share with the class what a polygon is? *Can you find the area of each of the POLYGONS you drew? *** 1
Can you construct.. 1. A triangle with an area of 18 cm 2? 2. A square with an area of 16 cm 2? 3. A parallelogram with an area of 24 cm 2? 4. A trapezoid with an area of 20 cm 2? 2
Unit 1 Regular Polygons 3
Essential Questions How are polygons used in our daily lives? How are perimeter and area related? 4
Main Ideas What is a polygon?; Constructing regular polygons; Axes of symmetry in regular polygons; Area and perimeter of regular polygons. Objective To be able to solve problems involving polygons! 5
Objective To be able to solve problems involving polygons! The adjacent drawing represents the floor of the new solarium in the Smith home. The perimeter of the solarium, in the shape of a regular hexagon, is 18 m. In the centre of the solarium, there is a rectangular ceramic decoration. The width of the rectangle is 1.5 m. The ratio between the width and the length of the rectangle is 1:2. Mrs. Smith wants to cover the rest of the floor with a carpet that is sold at $24 a square metre, all taxes included. 5 7 2 m 9 0 1 8 3 4 2 6 1.5 m How much will the carpet cost? 6
What is a POLYGON? A polygon is a closed figure that is constructed using 3 or more line segments. Is a circle a polygon? No, because it does not have line segments. 7
Polygon Terminology Polygon Concave polygon Convex polygon Exterior angle of a polygon Regular polygon Apothem Triangle Square Nonagon Heptagon Hendecagon Pentagon Decagon Octagon Dodecagon Hexagon 8
Connect all 4 points to make different polygons How many different polygons can you create? A C B A B ABDCA C D A ACBDA B C D D 9
Polygons like the one below are called intersecting. B A ABDCA C D In this unit we will study NON-INTERSECTING B polygons, like this: A ABCDA C D 10
Compare these polygons How are they alike? How are they different? Extend each side with a dotted line A polygon is considered CONVEX when the sides are extended and they do not pass through the interior. If the sides do pass through the interior it is a CONCAVE polygon. 11
Interior Angle > 180 Indicates a CONCAVE polygon! 12
What is a diagonal? A diagonal is a line joining 2 non-adjacent vertices (corners) of a polygon. 13
Total Number of Diagonals *page 11 table* # Sides or Vertices # Diagonals from each vertex Total # of diagonals 3 0 0 4 1 5 6 7 8 n 14
All polygons can be divided into triangles The TRIANGLE is the basic shape of all polygons. 2 Draw each diagonal from the same vertex. 1 3 1 2 15
Drawing diagonal to divide each of these polygons into triangles *page 11 table* Predict how many triangles a heptagon can be divided into Do you see a pattern? an octagon? # Sides 3 4 5 6 7 8 n # Triangles 1 16
Interior Angle The interior angles of a polygon are those angles at the vertex that are on the inside of the polygon. There is one interior angle at each vertex. 17
a Sum of the Interior Angles *page 11 table* b c # Sides # Triangles Sum of Interior Angles 3 1 1 x 180 a d e 4 2 2 x 180 5 3 3 x 180 = 360 = 540 b c f 6 4 4 x 180 7 5 5 x 180 8 6 6 x 180 = 720 = 900 = 1080 b a g d c h i e f 9 7 7 x 180 n n-2 (n-2) x 180 Sum of the interior angles of a polygon with n sides = (n-2) x 180 = 1260
Exterior Angle The sum of the exterior angles of a convex polygon is 360 105 100 75 80 105 + 100 + 80 + 75 = 360 19
Concept Attainment Compare and contrast columns A and B, think about the similarities and differences. 20
Concept Attainment 21
60 60 60 Regular Polygons *page 11 table* 90 A regular polygon has all sides and angles congruent. # Sides 3 4 5 6 7 8 n Sum Interior Angles 180 360 540 720 900 1080 (n-2)180 Measure of each interior < 60 90 108 120 128.57 135 (n-2)180 n 120 108 22
Regular Polygons & Central Angles *page 11 table* Central angles are formed in regular polygons by joining the vertices to the centre of the regular polygon. # Sides 3 4 5 6 7 8 9 10 n Central Angle Measure B. Murphy 23
# Sides Name # s n-2 Total Diagonals n(n-3) 2 Sum of Interior < s (n-2)180 Interior < Measure (n-2)180 n Central < Measure 360 n 4 square 2 2 360 90 90 5 pentagon 3 5 540 108 72 6 hexagon 4 9 720 120 60 7 heptagon 5 14 900 128.57 51.43 8 octagon 6 20 1080 135 45 9 nonagon 7 27 1260 140 40 10 decagon 8 35 1440 144 36 11 hendecagon 9 44 1620 147.27 32.73 12 dodecagon 10 54 1800 150 30 24
Constructing Regular Polygons We can construct any REGULAR POLYGON using a protractor and ruler. We need to know side length and interior angle in order to construct the polygon. Let s begin with a regular OCTAGON with a perimeter of 16 cm; 1. First draw a straight line of required length, let s construct an octagon with a P = 16 cm; SIDE LENGTH = 16 cm 8 sides = 2 cm A 2. Next use your protractor to measure the interior angle needed, use the formula: n 2 180 n 8 2 180 8 1080 8 = = = 135 B 3. Draw a line from point to new point, measuring the exact same length as AB 4. Repeat steps 2 and 3 until polygon is constructed. 25
Constructing Regular Polygons #1 Construct a regular heptagon with P = 14 cm. Show the number of diagonals. Verify #2 Construct a regular nonagon with a P= 22.5 cm. Show the number of triangles. Verify #3 Construct a regular decagon with a P = 30 cm. 26
a a+a+a+a+a+a = 6a 6 (a) = 6a How would you calculate the perimeter of each of these regular polygons? a+a+a = 3a 3 (a) = 3a Pentagon Hexagon Octagon Decagon a Side length Perimeter a b c d 27
Area of Regular Polygons Apothem, is a line segment dropped from the centre of a regular polygon and it is perpendicular to any one of the sides. It is usually denoted by the letter a O a Can you suggest a formula to calculate the area of this pentagon? Area of a regular polygon = P (a) 2 28
Area of Regular Polygons Area of a regular polygon = P (a) 2 1. Calculate the area of each of these regular polygons. 12cm 2. a = 7cm a = 5.5cm 29
Different ways of using the formula Looking for AREA: A = P(a) 2 Looking for PERIMETER: P = 2(A) a Looking for APOTHEM: a = 2 (A) P 30
Find the missing term: Decagon A = 123.2 m 2 a = 6.16 m Side length =? Octagon A = 77.28 cm 2 P = 32 cm Apothem=? 31