Page 1 CCM6+ Unit 11 Angle Relationships, Area, Perimeter/Circumference 15-16 UNIT 11 Angle Relationships, Area, and Perimeter/Circumference 2016 2017 CCM6+ Name: Math Teacher: Projected Test Date: Unit 11 Vocabulary.........2 Geometric Vocabulary and Angle Relationships..... 3-11 Perimeter and Area of Polygons.........12-23 Area of Composite Shapes...24-29 Formulas and Area....30-32 Area and Circumference of Circles.....33-38 Mixed Area Word Problems.. 39-45 Area and Perimeter on a Coordinate Plane...46-48 Unit Review/Study Guide... 49-53 1
Page 2 CCM6+ Unit 11 Angle Relationships, Area, Perimeter/Circumference 15-16 perimeter area polygon regular polygon rectangle triangle acute triangle obtuse triangle right triangle hypotenuse equilateral triangle isosceles triangle scalene triangle parallelogram circle circumference center chord diameter radius pi angle VOCABULARY the measure around an object the amount of space inside a figure a closed plane figure formed by 3 or more line segments that intersect only at their endpoints. a figure that has all equivalent sides and angles. a parallelogram with four right angles a 3-sided polygon a triangle with only acute angles a triangle with one obtuse angle a triangle with one right angle the longest side of a right triangle a triangle in which all sides are equal a triangle with at least two congruent sides a triangle with no congruent sides a four sided figure with opposite sides that are equal and parallel the set of all points in a plane that are the same distance from a given point called the center the distance around a circle the point inside a circle that is the same distance from all points on the circle a line segment with both endpoints on the circle a line segment that passes through the center of a circle and has endpoints on the circle, or the length of that segment a line segment with one endpoint at the center of a circle and the other endpoint on the circle, or the length of that segment the ratio of the circumference of a circle to the length of its diameter; π 3.14 or formed by two rays that begin at the same point or share the same endpoint congruent angles Angles that have the same measure. (Holt p. 331) acute angle measures less than 90 obtuse angle measures more than 90 right angle measures exactly 90 straight angle measures exactly 180 adjacent angle Angles in the same plane that have a common vertex and a common side. (Holt p. 332) vertical angle A pair of opposite congruent angles formed by interesting lines. (Holt p. 332) complementary angle two angles whose measures have a sum of 90 supplementary angle two angles whose measures have a sum of 180 2
Page 3 CCM6+ Unit 11 Angle Relationships, Area, Perimeter/Circumference 15-16 WARMUP: Classifying Angles and Triangles For each picture, tell anything you already know. Picture What I know What I learned 3
Page 4 CCM6+ Unit 11 Angle Relationships, Area, Perimeter/Circumference 15-16 A 10 in 10 in B 10 in C All three angles in a triangle 4
Page 5 CCM6+ Unit 11 Angle Relationships, Area, Perimeter/Circumference 15-16 ANGLE Types.5 types What are angles made from? Two connected at a point. Acute angles measure less than. Sketch 3 acute angles below. Right angles measure exactly. The symbol often used to show a right angle is:. Sketch a right angle below. Obtuse angles measure over but under. Sketch 3 obtuse angles below. Straight angles measure exactly. Sketch a straight angle below. Reflex angles measure over but under. Sketch a reflex angle below. ANGLE RELATIONSHIPS Two or more angles are required to use these words. Adjacent share one ray side by side. Angles 1 and 2 are adjacent. Angles c and d are not adjacent. 5
Page 6 CCM6+ Unit 11 Angle Relationships, Area, Perimeter/Circumference 15-16 Complementary two or more angles that add up to. One is the complement of the other. Complementary angles do not have to be adjacent. Adjacent complementary angles. Nonadjacent Complementary Angles. 3 Complementary Angles. Supplementary two or more angles that add up to. One is the supplement of the other. Supplementary angles do not have to be adjacent. Vertical two angles created from intersecting lines that are opposite from the vertex. They are. Vertical angles can never be adjacent, since they never share a ray. A s are vertical and B s are vertical. A s have same measure (acute). Angles 2 and 4 are vertical. Angles 1 and 3 are vertical. B s have same measure (obtuse). Angle 1 = Angle 3. Angle 2 = Angle 4. A + B = 180 (supplementary) Angle 1 + Angle 2 = 180 (supplementary) 6
Page 7 CCM6+ Unit 11 Angle Relationships, Area, Perimeter/Circumference 15-16 Angles in a triangle add up to. So if you know two measures, you can find the measure of the missing interior angle. Find the measures of the missing triangle interior angles. Practice Find the missing angle measures based on the relationships above. 1) 2) 3) Find the measures of angles x, y, and z. 4) Find the measure of the missing angle a. 4) Find the measures of all missing angles. Write them onto the diagram. 7
Page 8 CCM6+ Unit 11 Angle Relationships, Area, Perimeter/Circumference 15-16 4. Write and solve an equation to find the measure of angle x. 8
Page 9 CCM6+ Unit 11 Angle Relationships, Area, Perimeter/Circumference 15-16 5. Find the measure of angle x. 6. Find the measure of angle b. Note: Not drawn to scale. 7. Find the measure of all missing angles. Draw them onto the diagram. 9
Page 10 CCM6+ Unit 11 Angle Relationships, Area, Perimeter/Circumference 15-16 10
Page 11 CCM6+ Unit 11 Angle Relationships, Area, Perimeter/Circumference 15-16 11
Page 12 CCM6+ Unit 11 Angle Relationships, Area, Perimeter/Circumference 15-16 Perimeter and Area WARMUP: Answer the two questions and fill in the chart below. Complete this page and the next two. Mr. Bill s backyard is in the shape of a rectangle. It took him 600 feet of fence to enclose his back yard. If the length of the yard is twice as long as the width, what are the dimensions of Mr. Bill s yard? The Brown family has a square back yard with an area of 25 meters squared. They need to put a fence around it for their dog. How long will the fence be? Fill in anything you know: SHAPE How to find perimeter How to find area square rectangle triangle parallelogram Now, complete all you can on the next page. 12
Page 13 CCM6+ Unit 11 Angle Relationships, Area, Perimeter/Circumference 15-16 13
Page 14 CCM6+ Unit 11 Angle Relationships, Area, Perimeter/Circumference 15-16 PERIMETER REVIEW What is it? How do I calculate it? PERIMETER AREA Find the Perimeter of each shape below. Find the length of the missing side if given the Perimeter of the whole shape. If the perimeter of a regular hexagon is 30cm, what is the length of one side? 14
Page 15 CCM6+ Unit 11 Angle Relationships, Area, Perimeter/Circumference 15-16 Perimeter Review, Area of Polygons The Relationship between Rectangles and Triangles 15
Page 16 CCM6+ Unit 11 Angle Relationships, Area, Perimeter/Circumference 15-16 Calculating Area of Squares/Rectangles/Parallelograms A parallelogram is just a in disguise! SHOW IT! Area formulas you need to KNOW: Shape Formula Example Solved Together Your Turn Square AREA= AREA= Rectangle AREA= AREA= Parallelogram AREA= AREA= 16
Page 17 CCM6+ Unit 11 Angle Relationships, Area, Perimeter/Circumference 15-16 WHAT BIG IDEA DO YOU NOTICE WITH TRIANGLES? 17
Page 18 CCM6+ Unit 11 Angle Relationships, Area, Perimeter/Circumference 15-16 On your calculator, type APPS Choose AreaForm Press any key twice Choose #1 Choose 3: Parallelogram After it defines the parallelogram, click the WINDOW key to see the area formula. When it finishes telling the formula, click the GRAPH key to see Why? Click GRAPH again. Do it again with #4: Triangle and check out #5 Trapezoid. **You don t have to know trapezoids. Here are two right triangles: Here are two non-right triangles: What shape do these create together? What shape do these create together? EVERY TRIANGLE DOUBLED MAKES EITHER A or a. Since a is really a tilted (same area), the area of a triangle is always of the area of a or a. FORMULA FOR THE AREA OF A TRIANGLE: A = ( ) Find the Area of each shape: 18
Page 19 CCM6+ Unit 11 Angle Relationships, Area, Perimeter/Circumference 15-16 REVIEW of AREA of TRIANGLES and PARALLELOGRAMS How do I find the area of a parallelogram? Area = base x height Since a parallelogram is similar to a rectangle, the base and height are relative to the length and width. * Be careful to measure the height and NOT the length of the slanted side. Parallelogram and its relation to a rectangle: (teachers: Demonstrate this using a piece of paper) Notice that when the outside piece is cut off and pasted to the other side of the parallelogram the polygon that is formed is a rectangle. Practice: H= 1.25 ft Area = bh Area = 2.5 x 1.25 Area = 3.125 square ft B = 2.5 ft * remember to label with square units, because this is a two dimensional figure. Practice: 4.5 ft 4ft 12 ft Area = bh Area = 12 x 4 Area = 48 square feet. 19
Page 20 CCM6+ Unit 11 Angle Relationships, Area, Perimeter/Circumference 15-16 Practice: A parallelogram has these measures: Base = 4.1 inches Height = 2.2 inches Answer: 9.02 square inches Practice: A parallelogram has these measures: Base = 4 yd Height = 9 yd Answer: 36 How do I find the area of a triangle? Look at this rectangle When I split the rectangle in half what shapes are formed? Two triangles are formed when a rectangle is split in half. So, half of a rectangle is a triangle. So, half of the area of a rectangle is a triangle. Therefore, Area of a triangle = 1 bh 2 NOTE: Multiplying by 2 1 is the same as dividing by 2. You bh may see or use the formula like this: 2 Example: Area = ½ bh H = 10.5 m Area = 0.5 x 16.8 x 10.5 = B= 16.8 m Area = 88.2 sq. m or 88.2 m 2 *Remember this is still a two dimensional figure therefore your answer should be labeled in square units. 20
Page 21 CCM6+ Unit 11 Angle Relationships, Area, Perimeter/Circumference 15-16 Practice: Area = ½ bh H= 11 cm B = 6 cm Practice: The base of a triangle is 3 m and the height is 8 sq. m. What is the area? Start by writing the formula, then substituting for what you know. A = Practice: The base of a triangle is 8 cm and the area is 32 cm. What is the height of the triangle? (You will have to rearrange the formula) Start by writing the formula, then substituting for what you know. A = 21
HOMEWORK Day 2 Page 22 CCM6+ Unit 11 Angle Relationships, Area, Perimeter/Circumference 15-16 22
Page 23 CCM6+ Unit 11 Angle Relationships, Area, Perimeter/Circumference 15-16 Area of Trapezoids On your TI-73, use the AREAFORM application and watch the area formula for trapezoids. Write what you discovered in the space below. 23
Page 24 CCM6+ Unit 11 Angle Relationships, Area, Perimeter/Circumference 15-16 Area of Composite Shapes WARMUP: What do we do if the shapes are MIXED? Mixed shapes are called COMPOSITE shapes. To find the area you have to. Total Area: square units (Hint: Get a ruler!) Find the area of the irregular polygon below. Measurements have been provided for you this time. 5 cm 4 cm 8 cm 2 cm 8 cm 4 cm 2 cm 10 cm 5 cm 24
Page 25 CCM6+ Unit 11 Angle Relationships, Area, Perimeter/Circumference 15-16 Practice DRAW IT! 1. Find the area of a right triangle with a base length of three units, a height of four units, and a hypotenuse of 5. HINT: the hypotenuse is always the biggest side and isn t part of the right angle. 2. Find the area of the trapezoid shown below using the formulas for rectangles and triangles. 12 3 7 3. A rectangle measures 3 inches by 4 inches. If the lengths of each side double, what is the effect on the area? 25
Page 26 CCM6+ Unit 11 Angle Relationships, Area, Perimeter/Circumference 15-16 4. The lengths of the sides of a bulletin board are 4 feet by 3 feet. How many index cards measuring 4 inches by 6 inches would be needed to cover the board? 5. The sixth grade class at Hernandez School is building a giant wooden H for their school. The H will be 10 feet tall and 10 feet wide and the thickness of the block letter will be 2.5 feet. 1. How large will the H be if measured in square feet? 2. The truck that will be used to bring the wood from the lumberyard to the school can only hold a piece of wood that is 60 inches by 60 inches. What pieces of wood (how many and which dimensions) will need to be bought to complete the project? 6. A border that is 2 ft wide surrounds a rectangular flowerbed 3 ft by 4 ft. What is the area of the border? 26
Page 27 CCM6+ Unit 11 Angle Relationships, Area, Perimeter/Circumference 15-16 Area of Composite Shapes HOMEWORK 27
Page 28 CCM6+ Unit 11 Angle Relationships, Area, Perimeter/Circumference 15-16 (Still Homework Day 3) 28
Page 29 CCM6+ Unit 11 Angle Relationships, Area, Perimeter/Circumference 15-16 Area Formulas and Parts of Equations/Expressions Warm-up 29
Page 30 CCM6+ Unit 11 Angle Relationships, Area, Perimeter/Circumference 15-16 Today s Big Idea: Formulas 1. The formula for volume of a rectangle prism is V = lwh. Find the height if you have the following information: V = 240 in 2 ; l = 12in; w = 4in 2. How wide is a rectangular strip of land with length ¾ mile and area ½ square mile? 3. 30
Page 31 CCM6+ Unit 11 Angle Relationships, Area, Perimeter/Circumference 15-16 31
Page 32 CCM6+ Unit 11 Angle Relationships, Area, Perimeter/Circumference 15-16 Formulas Practice Objectives: 6.EE2c: Evaluate expressions by substituting given values for variables and use formula to solve the problems. Practice 1. The formula that is used to convert Fahrenheit (F) to Celsius (C) is C = 5 (F 32) 9. Convert 77 F to degrees in Celsius. Steps for writing an algebraic expression for word problems: 1. Write a verbal phrase 2. Identify the operations is used 3. Define variables 4. Write expressions 2. Find the total area of a rectangle tile using the formula A=lw with the length is 6in, w is 8in. 3. Find the volume of a cubical packing box using the formula V = s 3 if the side length is 2ft. 4. Ms. Li want to make a rectangular garden. The length of the garden is 8ft and width of the garden is 7ft. Use the formula P = 2l + 2w to find the perimeter of the garden. 5. The speed limit along a particular highway increased from 55 mph to 65 mph. How much time will be saved on a 100-mile trip? Hint: d = r t 6. The Party Shoppe is advertising a special sale on balloons. They have two sizes available, 9-inch diameter and 12-inch diameter. Using the formula for the volume of a sphere, V = πr 3, determine the amount of helium needed to fill each of the balloons. Write your answer in terms of π. 32
Page 33 CCM6+ Unit 11 Angle Relationships, Area, Perimeter/Circumference 15-16 Circles 33
Page 34 CCM6+ Unit 11 Angle Relationships, Area, Perimeter/Circumference 15-16 34
Page 35 CCM6+ Unit 11 Angle Relationships, Area, Perimeter/Circumference 15-16 THE BASICS of CIRCLES Circumference = The distance a circle. Formula: C = Area = The space a circle. Use a TI-73 and go to APPS and choose AreaForm. Press any key twice to get to the main menu. Choose 1: Definitions and Formulas Choose 6: Circle When it finishes defining a circle, hit the WINDOW key for AREA. Hit GRAPH for Why?. Keep hitting GRAPH for Why? to see the reason for the formula. What shape did they make out of a circle? What is the area formula of that shape? Draw your discovery here. Label the dimensions! Height= Base = A= or A = 35
Now, practice! Page 36 CCM6+ Unit 11 Angle Relationships, Area, Perimeter/Circumference 15-16 CIRCLE Radius Diameter Circumference Area 36
Page 37 CCM6+ Unit 11 Angle Relationships, Area, Perimeter/Circumference 15-16 Circumference and Area of Circles Practice Name: Date: 1. The seventh grade class is building a mini-golf game for the school carnival. The end of the putting green will be a circle. If the circle is 10 feet in diameter, how many square feet of grass carpet will they need to buy to cover the circle? How might someone communicate this information to the salesperson to make sure he receives a piece of carpet that is the correct size? Use 3.14 for pi. 2. If a circle is cut from a square piece of plywood, how much plywood would be left over? 3. What is the perimeter of the inside of the track? 37
Page 38 CCM6+ Unit 11 Angle Relationships, Area, Perimeter/Circumference 15-16 WARMUP Problem Solving Remember π 3.14 or π 22 7 1. A coffee cup has a diameter of 3 1 inches. What is its circumference? 2 2. A circle has a diameter of 4 3 inches. What is the circumference? Round your answer to 10 the nearest tenth. 3. What is the area of the circle if its radius is 2 cm? 4. What is the area of a circle that has a diameter of 6 cm? 5. If the area of a circle is 200.96 m 2, then what is its diameter? 38
Page 39 CCM6+ Unit 11 Angle Relationships, Area, Perimeter/Circumference 15-16 Shape within a Shape (with Circles) SET UP YOUR WORK HERE But use a CALCULATOR to solve! Answer choices are on p. 41. 1) Jason and his brother purchased a round rug to lay in their living room. The living room is 13ft x 11ft, and the rug is 4 feet in diameter. What is the area of the room NOT covered by the rug? 2) What is the area of the shaded region if the length of this square is 2.2 cm? 3) The room for the wedding is 30 ft by 32 ft. There are 5 round tables for guests. Each table has a diameter of 6 ft. What is the area of room that is available for dancing and walking? 4) What is the area of the shaded region if the radius of this circle is 3.4 in? 39
Page 40 CCM6+ Unit 11 Angle Relationships, Area, Perimeter/Circumference 15-16 5) Rachel put her coffee table on top of a circular rug. The circular rug has a diameter of 4 feet. The round coffee table has a diameter of 3 feet. How much of the rug is NOT covered by the coffee table? 6) The triangle s base is 2.5 cm and is equal to the radius of the circle. What is the area of the shaded region? 7) A circular table with a diameter of 4 feet has four circular place mats on it, each with a diameter of 6 inches. What is the area of the table in inches, NOT covered with place mats? 8) What is the area of the shaded region if the radius of the circle is 10.4 ft? 40
Page 41 CCM6+ Unit 11 Angle Relationships, Area, Perimeter/Circumference 15-16 Shape within a Shape Game Cards 9) Calculate the area of the shaded section in the picture below: 15 cm 10) Mary s father put a garden in their backyard that had an area of 5 ft. by 9 ft. He put a sidewalk around the garden that had an area of 7 ft. by 12 ft. What is the area of the sidewalk around the garden? 9cm 4 cm 9 cm 11) Calculate the area of the shaded section in the picture below: 12 yd 18 yd. The dimensions of the inner polygon are 3 yd. by 9 yd. 12) The area of a local school is 3,844 sq. meters. When they built the school they put a sidewalk around the school. The dimensions of the rectangle formed by the outer edge of the sidewalk are 72 meters by 70 meters. What is the area of the space between the school and sidewalk? 41
Page 42 CCM6+ Unit 11 Angle Relationships, Area, Perimeter/Circumference 15-16 13) Calculate the area of the shaded section in the picture below: The dimensions of the inner polygon are 8cm 12 cm by 5 cm 14) Bob built his very own lemonade stand in front of his house. His lemonade stand was 8 feet by 12 feet. He decided that he needed to make it look nicer by planting flowers all the way around the stand. The area of the rectangle formed around the planted flowers was 130 sq. feet. How much space was there between his lemonade stand and 11 cm the flowers? 15) Calculate the area of the shaded section in the picture below: 15 ft. 16) Regulation NCAA basketball courts have dimensions of 50 feet by 94 feet. There are chairs around the entire court that make up an area of 56 12 cm The dimensions of the inner polygon are 3ft by 6 ft feet by 100 feet. How much space is there just for the chairs? 42
Page 43 CCM6+ Unit 11 Angle Relationships, Area, Perimeter/Circumference 15-16 ANSWERS for SHAPE WITHIN A SHAPE: 900 sq ft 34 sq ft 92 sq cm 189 sq yd 39 sq ft 162 sq. ft 1196 sq. m 5.495sq. ft 216.32 sq ft 1695.6 sq in 818.7 sq. ft 9.9416 sq. in 3.7994 sq. cm 130.44 sq. ft 16.5 sq. cm 99 sq cm 43
Page 44 CCM6+ Unit 11 Angle Relationships, Area, Perimeter/Circumference 15-16 44
Page 45 CCM6+ Unit 11 Angle Relationships, Area, Perimeter/Circumference 15-16 45
Page 46 CCM6+ Unit 11 Angle Relationships, Area, Perimeter/Circumference 15-16 Graph figure PQRS: P(-4, 3), Q(10, 3), R(10, -3), S(-4, -3). Determine the area and perimeter of the figure. Give the coordinates of a figure that has a perimeter half that of figure PQRS. Give the coordinates of a triangle that has an area half that of figure PQRS. Graph rectangle MNOP : M ( 4, 3), N(10, 3), O(4, 7), P(10, 7). Determine the perimeter and area of the figure. Give the coordinates for rectangle QRST that has the same area, but a different perimeter. 10 8 6 4 2-10 -8-6 -4-2 2 4 6 8 10-2 -4-6 -8-10 10 8 6 4 2-10 -8-6 -4-2 2 4 6 8 10-2 -4-6 -8-10 ABC ). Graph triangle : A ( 4,9), B(1,3), C(8,3 Determine the area of the triangle. Give the coordinates for a triangle DEF that has an area twice that of triangle ABC. 10 8 6 4 2-10 -8-6 -4-2 2 4 6 8 10-2 -4-6 -8-10 46
Page 47 CCM6+ Unit 11 Angle Relationships, Area, Perimeter/Circumference 15-16 47
Page 48 CCM6+ Unit 11 Angle Relationships, Area, Perimeter/Circumference 15-16 48
Page 49 CCM6+ Unit 11 Angle Relationships, Area, Perimeter/Circumference 15-16 CCM6+ UNIT 11 STUDY GUIDE PERIMETER AND AREA Tell how to calculate the following. Write the formula if there is a formula! 1. Perimeter 2. Area of a square 3. Area of a rectangle 4. Area of a parallelogram 5. Area of a triangle What is different about the triangle formula? How will you remember this? 6. Area of mixed shapes what do you do? What is tricky? 13. Find the perimeter of this rectangle. (an algebraic expression) 49
Page 50 CCM6+ Unit 11 Angle Relationships, Area, Perimeter/Circumference 15-16 WORD PROBLEMS and FORMULAS 21) The perimeter of a rectangle is 12. Determine a possible length and width, then calculate a possible area for that rectangle. 22) A rectangular photo is 5 inches long and 2 inches wide. Jimmy wants to enlarge the photo by doubling its length and width. How many inches of wood will he need to make a frame for the enlarged photo? 23) A figure is formed by a square and a triangle. Its total area is 32.5 m 2. The area of the triangle is 7.5 m 2. What is the length of each side of the square? a) 5 meters b) 25 meters c) 15 meters d) 16.25 meters 50
Page 51 CCM6+ Unit 11 Angle Relationships, Area, Perimeter/Circumference 15-16 24) A rectangle is formed by two congruent right triangles. The area of each triangle is 6 in 2. If each side of the rectangle is a whole number of inches, which of these could NOT be its perimeter? a) 26 inches b) 24 inches c) 16 inches d) 14 inches 25) The volume of a cube is found with the formula V=s 3 where the side length is represented by s. If the side length is 1 1 inches, what is the volume of the cube? 2 26) The perimeter of a rectangle is 20 ft 2. If the length is 5 ft, what is the AREA of the rectangle? For each problem: Plot the ordered pairs in the coordinate plane given Find the perimeter of the figure Find the area of the figure Find the distance between each point by using the absolute value method. 27. G (-4, 5) H (5, 5) I (-4, -5) J (5, -5) Perimeter of GHIJ: Area of GHIJ: 51
Page 52 CCM6+ Unit 11 Angle Relationships, Area, Perimeter/Circumference 15-16 28. This figure is a four sided polygon. Before finding the area and perimeter find the missing point. R (-2, 2) S (4, 2) T (, ) U ( -2, -3) Perimeter of RSTU: Area of RSTU: What was the fourth vertex? How did you find the length for each side of the figure? Find the area of the shaded region for each figure below. 29. 30. 52
Page 53 CCM6+ Unit 11 Angle Relationships, Area, Perimeter/Circumference 15-16 31. 32. 33. Find the Area and Perimeter. 53