Polygons and Angles. Polygons and Angles. Solutions. Curriculum Ready.
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1 Polygons and Angles Polygons and Angles Curriculum Ready
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3 Basics Page 3 questions 1. Circle the shapes that are polygons: 2. Name these polygons (based on the number of sides) and state whether each is regular or irregular: It has 8 sides therefore it is an octagon. Since, all the sides are the same length indicated by the dashes it is regular. This shape has 5 sides and is therefore a pentagon. However, since all the sides are not of equal length, indicated by the two different types of dashes (single dash and double dashes), the shape is irregular. This shape has four sides and is therefore a quadrilateral. However, since only all the sides are different lengths the shape is irregular. The shape has three sides each of different lengths. Because of this, it is a triangle which is irregular. 100% Polygons & Angles 1
4 Basics Page 3 questions This shape has 12 sides thereby making it a dodecagon. However, since the lengths are different, the shape is irregular. Since this shape has 5 unequal sides it is an irregular pentagon. This shape has four equal sides making it a regular quadrilateral. The name we give to a regular quadrilateral is a square. This shape has 11 sides all of varying length and so is an irregular undecagon. 3. Name these polygons (based on the number of sides) and state whether they are conve or concave. This shape has 4 sides making it a quadrilateral. Since all the interior angles are less than then it is considered conve. This shape has 3 sides, so it is a triangle. Since all the interior angles are less than it is considered conve. This polygon has 10 sides making it a decagon but since five interior angles are greater than it is concave % Polygons & Angles
5 Basics Page 3 questions This shape has 7 sides making it a heptagon but it has two interior angles greater than thereby making it concave. This polygon has 8 sides making it an octagon but since it has interior angles greater than it is concave. This polygon has 3 sides and therefore is a triangle. All interior angles are less than making it conve. This polygon has 4 sides making it a quadrilateral. Since one interior angle is greater than, it is concave. This shape has 9 sides so it is called a nonagon. It has one interior angle greater than and is therefore concave. 100% Polygons & Angles 3
6 Knowing More Page 5 questions 1. Use the circle below to draw an equilateral triangle: 120c 120c 120c A regular equilateral triangle has 3 sides. Therefore, 360 ' n to find the size of the angle between each radius. ` 360' 3 120c 2. Use the circle below to draw a regular pentagon: 72c 72c 72c 72c 72c A regular pentagon has 5 sides. Therefore, 360 ' n to find the size between each radius. ` 360' 5 72c 3. Use the circle below to draw a regular octagon: 45c 45c 45c 45c 45c 45c 45c 45c A regular octagon has 8 sides. Therefore, 360 ' n to find the size between each radius. ` 360' 8 45c 4 100% Polygons & Angles
7 Using Our Knowledge Page 8 questions 1. Find the value of in each of the following: a Since the polygon is a triangle the sum of the interior angles. 60c + 95c+ 60c -95c- 60c 95c 25c b 39c 123c Since the polygon is a quadrilateral the sum of the interior angles 360c c+ 123c+ 39c 360c 118c 360c-118c-123c-39c 80c c 110c This question involves finding the supplementary angles given the following information. Thesupplementangle of 62cis- 62c 118c. Thesupplementary angleof110cis - 110c 70c. Since the polygon is a quadrilateral the interior angles sum to 360c. 62c + 90c+ 118c+ 70c 360c 360c-90c-118c-70c 82c d 30c The given shape is a triangle. The sides with dashes indicate that they have the same length and therefore have the same angle formed at the base c 2-30c 2 150c 150c ' 2 75c 100% Polygons & Angles 5
8 Using Our Knowledge Page 8 questions e 2 This polygon has 4 sides making it a quadrilateral so the interior angles sum to 360c. Notice that 120c is an eterior angle; we can find the interior angle above it by subtracting 120c from 360c therefore making the interior angle 240c c+ 48c 360c 48c 120c 3 360c-240c- 48c 3 72c 72c ' 3 24c f 2 6 Since the shape is a quadrilateral the sum of the interior angles is 360c c c 3 360c ' c Page 9 questions 2. Divide this heagon into 4 triangles by drawing diagonals from a common verte % Polygons & Angles
9 Using Our Knowledge Page 9 questions 3. Complete the following table for polygons and with different sides: Number of sides ^nh Number of triangles formed ^n - 2h Sum of interior angles ^n - 2h # ^4-2h# 360c (5-2) # 540c (6-2) # 720c (7-2) # 900c ( 8-2) # 1080c ( 9-2) # 1260c ( 10-2) # 1440c ( 11-2) # 1620c ( 12-2) # 1800c 4. Find the sum of the interior angles of a polygon with: a 17 sides ( n -2)# (17-2)# 15 # 2700c b 27 sides ( n -2)# ( 27-2)# 25 # 4500c Page 10 questions 5. Answer these questions about the following polygon: a How many sides does this polygon have? The polygon has 10 sides. 155c 162c 148c 146c 160c b What is the sum of the interior angles of this polygon? ( n -2)# (10-2)# 8# 1440c 167c 130c 134c 126c 100% Polygons & Angles 7
10 Using Our Knowledge Page 10 questions c Find the size of c+ 130c+ 126c+ 134c+ 160c+ 146c+ 148c+ 162c+ 155c 1440c 1440c-167c-130c-126c-134c-160c-146c-148c-162c-155c 112c 6. Find the number of sides of a polygon (n) which has a sum of interior angles of: a 2880c b 3600c ( n - 2) # 2880c n c n n n 18 ( n - 2) # 3600c n c n n n What is the size of each angle in each of the following polygons? a Regular heptagon b Regular undecagon ( n - 2) # n where n 7 ( n - 2) # n where n 11 ( 7 2) # c 7 5# 7 900c c ( 1d.p.) ( 11-2) # 11 9# c c ( 1d.p.) 8 100% Polygons & Angles
11 Thinking More Page 14 questions 1. Use the eterior angles of a triangle to find in the following diagrams: a b 33c 50c+ 70c 120c 80c + 33c 80c 80c-33c 47c 70c 50c c 137c 43c+ 90c 133c - 137c 43c d 2c 132c c 3 132c 132c ' 3 44c e A B 2 C 17c D E 110c F It is easiest to split this up into parts before completing this question. In order to find it is first necessary to find + BDE. + BDE 110c-17c + BDE 93c Once + BDE is known it is possible to find c 3 93c 93c ' 3 31c 100% Polygons & Angles 9
12 Thinking More Page 15 questions 2. Find in the following diagram: a 55c 17c 66c 62c 139c 86c Note the angle that is 139cis an interior angle. So the eterior angle is - 139c 41c. + 86c+ 62c+ 17c+ 55c+ 66c+ 41c 360c 360c-86c-62c-17c-55c-66c- 41c 33c b 78c 125c 2 130c 158c 27c Again, notice that there are three interior angles in this question. It is necessary to find the eterior angle to answer the question c 22c - 125c 55c - 130c 50c c+ 22c+ 55c+ 50c+ 27c 360c 4 360c-22c-78c-55c-50c-27c 4 128c 128c ' 4 32c % Polygons & Angles
13 Thinking More Page 16 questions 3. What size are the eterior angles of a regular quadrilateral? Sizeof eterior angle 360c n 360c 4 90c The eterior angles of a regular quadrilateral equate to 90ceach. 4. What size are the eterior angles of a regular decagon? Sizeof eterior angle 360c n 360c 10 36c The eterior angles of a regular decagon equate to 36ceach. 5. For a regular polygon with 15 sides: a What are the sizes of the eterior angles? Sizeof eterior angle 360c n 360c 15 24c The eterior angles of a regular decagon equate to 24ceach. b What size are the interior angles? n 15 ( 15 ) Size of interior angle - 2 # # c c 100% Polygons & Angles 11
14 Thinking More Page 16 questions c What is the sum of the interior angles? Sumofinteriorangles ( n -2)# ( 15-2)# 13 # 2340c 6. A regular polygon has eterior angles which are 18c. a How many sides does this polygon have? 360c n 360c 18 n n 360c ' 18 n 20 b What size are the interior angles? n 20 ( 20 2) 180 Size of interior angle - # c # c c c What is the sum of the interior angles? Sumofinteriorangle ( n -2)# ( 20-2)# 18 # 3240c % Polygons & Angles
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