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?
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1 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
2 Unit 1 Regular Polygons 2
3 Essential Questions How are polygons used in our daily lives? How are perimeter and area related? 3
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! B. Murphy 4
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 m m How much will the carpet cost? 5
6 What is a POLYGON? A polygon is a closed figure that is constructed using line segments. Is a circle a polygon? 6
7 Polygon Terminology Square Polygon Concave polygon Convex polygon Exterior angle of a polygon Regular polygon Apothem Triangle Pentagon Hexagon Heptagon Octagon Nonagon Decagon Hendecagon Dodecagon B. Murphy 7
8 Connect these points to make different polygons How many different polygons can you create? A C B B D B A C A C D D 8
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 B. Murphy 9 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. B. Murphy 10
11 Interior Angle > 180 Indicates a CONCAVE polygon! B. Murphy 11
12 What is a diagonal? A diagonal is a line joining 2 non-adjacent vertices (corners) of a polygon. 12
13 Total Number of Diagonals *page 136 table* # Sides or Vertices # Diagonals from each vertex Total # of diagonals n 13
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 B. Murphy 14
15 Drawing diagonal to divide each of these polygons into triangles *page 136 table* Predict how many triangles a heptagon can be divided into Do you see a pattern? an octagon? # Sides n # Triangles
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. for a polygon with n sides, there are n vertices and n interior angles. B. Murphy 16
17 a Sum of the Interior Angles *page 136 table* b c # Sides # Triangles Sum of Interior Angles x 180 a d e x b c f b a g d c h i e f 9 n Sum of the interior angles of a polygon with n sides = B. Murphy 17
18 Polygon Properties Triangles are the basic shape of all polygons; all polygons can be divided into n - 2 triangles, where n is the # of sides. The sum of the interior angles of a polygon is equal to the number of sides minus two times 180 The sum of the exterior angles of a convex polygon is = 360 B. Murphy 18
19 Concept Attainment Compare and contrast columns A and B, think about the similarities and differences. B. Murphy 19
20 Concept Attainment B. Murphy 20
21 Regular Polygons 90 A regular polygon has all sides and angles congruent. # Sides n Sum Interior Angles Measure of each interior < B. Murphy 21
22 Regular Polygons & Central Angles Central angles are formed in regular polygons by joining the vertices to the centre of the regular polygon. # Sides n Central Angle Measure 120 B. Murphy 22
23 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 SUM OF I.A. = = = = 135 n Draw a line from point to new point, measuring the exact same length as AB B 4. Repeat steps 2 and 3 until polygon is constructed. B. Murphy 23
24 Constructing Regular Polygons #1 Construct a regular heptagon with P = 14 cm. #2 Construct a regular nonagon with a P= 22.5 cm. #3 Construct a regular decagon with a P = 30 cm. 24
25 a How would you calculate the perimeter of each of these regular polygons? a Pentagon Hexagon Octagon Decagon Side length Perimeter a b c d A regular polygon with n sides of b units has a perimeter of nb units. B. Murphy 25
26 Area of Regular Polygons How would we find the area of this triangle? 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 a O (b xh)/2 b x h 2 Can you suggest a formula to calculate the area of this pentagon? Area of a regular polygon with n sides = B. Murphy 26
27 Area of Regular Polygons Area of a regular polygon with n sides = P a 2 1. Calculate the area of each of these regular polygons. 12cm 2. a = 7cm a = 5.5cm B. Murphy 27
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