# Page 1 CCM6+7+ UNIT 9 GEOMETRY 2D and 3D. Angle Relationships, Area, and Perimeter/Circumference Surface Area and Volume

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1 Page 1 CCM6+7+ UNIT 9 GEOMETRY 2D and 3D UNIT Angle Relationships, Area, and Perimeter/Circumference Surface Area and Volume CCM6+7+ Name: Math Teacher: Projected Test Date: MAIN CONCEPT(S) PAGE(S) Vocabulary 2-3 2D Perimeter 3-6 2D Area D Area of Composite Shapes D Area and Circumference of Circles D Inscribed and Composite Shapes D Graphing Shapes to find Area and Perimeter D shapes names, parts, nets, cross-sections D Surface Area Prisms D Surface Area Cylinders D Volume (prisms, cylinders, cones, pyramids, spheres) Fractional Edge Length and Problem Solving Study Guide

2 Page 2 CCM6+7+ UNIT 9 GEOMETRY 2D and 3D area area of a circle area of a parallelogram area of a rectangle area of a trapezoid area of a triangle center chord circle circumference cone cross-section cylinder diameter edge face hypotenuse net parallelogram perimeter pi polygon polyhedron prism pyramid radius rectangle rectangular prism regular polygon rhombus sphere square surface area trapezoid the amount of space inside a figure the number of square units contained within a circle, the area of a circle can be found using A=πr 2 and the units will be squared A = b x h A= l x w A = ½(b1+ b2)h A=1/2 x b x h 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 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; circumference can be found using the following formulas C=πd if given the diameter or C=2πr if given the radius a pyramid with a circular cross section the two-dimensional face that is the result of a three-dimensional shape being intersected by a plane. a 3-D figure with two circular ends a line segment that passes through the center of a circle and has endpoints on the circle, or the length of that segment the line segment along which two faces of a polyhedron intersect a flat surface of a polyhedron (a 3D figure) the longest side of a right triangle an arrangement of two-dimensional figures that can be folded to form a polyhedron (3-D figure); what you get if you unfold a shape a four sided figure with opposite sides that are equal and parallel the measure around an object the ratio of the circumference of a circle to the length of its diameter; π 3.14 or 22/7 a closed plane figure formed by 3 or more line segments that intersect only at their endpoints. three-dimensional figure whose surfaces, or faces, are all polygons a three dimensional figure with two parallel bases that are congruent polygons, and lateral faces that are parallelograms. A prism is named for the shape of its base. a polyhedron that has a polygon base and triangular lateral faces 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 a parallelogram with four right angles 3D figure where 6 faces are rectangles a figure that has all equivalent sides and angles. a parallelogram with four congruent sides a perfectly round 3-D figure a parallelogram with four right angles and four congruent sides the sum of the area of the faces of a 3D figure a quadrilateral with exactly one pair of parallel sides 2

3 Page 3 CCM6+7+ UNIT 9 GEOMETRY 2D and 3D triangle triangular prism vertices volume a 3-sided polygon a prism with bases that are triangles. a point where three or more edges intersect; the corners the number of cubic units needed to fill a given space 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 3

4 Page 4 CCM6+7+ UNIT 9 GEOMETRY 2D and 3D 4

5 PERIMETER REVIEW Page 5 CCM6+7+ UNIT 9 GEOMETRY 2D and 3D 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? 5

6 Page 6 CCM6+7+ UNIT 9 GEOMETRY 2D and 3D 6

7 Page 7 CCM6+7+ UNIT 9 GEOMETRY 2D and 3D Formulas of 2D shapes Area of a square or rectangle: A= On your calculator, hit APPS and AreaForm. Choose 1: DEFINITIONS & FORMULAS For each shape (Parallelogram, Triangle, Trapezoid) hit WINDOW to see the AREA formula and keep hitting GRAPH to see WHY? Record what you learned: PARALLELOGRAM: TRIANGLE: TRAPEZOID: 7

8 Page 8 CCM6+7+ UNIT 9 GEOMETRY 2D and 3D 8

9 Page 9 CCM6+7+ UNIT 9 GEOMETRY 2D and 3D 9

10 Page 10 CCM6+7+ UNIT 9 GEOMETRY 2D and 3D 10

11 Page 11 CCM6+7+ UNIT 9 GEOMETRY 2D and 3D 11

12 Page 12 CCM6+7+ UNIT 9 GEOMETRY 2D and 3D 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. 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 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. 12

13 Page 13 CCM6+7+ UNIT 9 GEOMETRY 2D and 3D 2. Find the area of the trapezoid shown below A rectangle measures 3 inches by 4 inches. If the lengths of each side double, what is the effect on the area? 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? 13

14 Page 14 CCM6+7+ UNIT 9 GEOMETRY 2D and 3D (Still Homework Day 3) 14

15 Page 15 CCM6+7+ UNIT 9 GEOMETRY 2D and 3D 15

16 Page 16 CCM6+7+ UNIT 9 GEOMETRY 2D and 3D Area Formulas and Parts of Equations/Expressions 16

17 Page 17 CCM6+7+ UNIT 9 GEOMETRY 2D and 3D Circles 17

18 Page 18 CCM6+7+ UNIT 9 GEOMETRY 2D and 3D 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 = Now, practice! 18

19 Page 19 CCM6+7+ UNIT 9 GEOMETRY 2D and 3D CIRCLE Radius Diameter Circumference Area 19

20 Page 20 CCM6+7+ UNIT 9 GEOMETRY 2D and 3D 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? 20

21 Problem Solving Remember π 3.14 or π 22 Page 21 CCM6+7+ UNIT 9 GEOMETRY 2D and 3D 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 m 2, then what is its diameter? 21

22 Page 22 CCM6+7+ UNIT 9 GEOMETRY 2D and 3D Notes 2D Irregular Figures Find the area of the shaded region of each of the following figures. 22

23 Page 23 CCM6+7+ UNIT 9 GEOMETRY 2D and 3D Shape within a Shape (with Circles) SET UP YOUR WORK HERE But use a CALCULATOR to solve! Answer choices are on p ) 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? 5) 6) Rachel put her coffee table on top of a circular rug. The circular rug has a 23

24 Page 24 CCM6+7+ UNIT 9 GEOMETRY 2D and 3D 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? 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? 24

25 Page 25 CCM6+7+ UNIT 9 GEOMETRY 2D and 3D 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? 25

26 Page 26 CCM6+7+ UNIT 9 GEOMETRY 2D and 3D 13) Calculate the area of the shaded section in the picture below: The dimensions of the inner 12 cm polygon are 8cm by 5 cm 11 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 the flowers? 15) Calculate the area of the shaded section in the picture below: 12 cm 15 ft. The dimensions of the inner polygon are 3ft by 6 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 feet by 100 feet. How much space is there just for the chairs? 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 sq ft sq in sq. ft sq. in sq. cm sq. ft 16.5 sq. cm 99 sq cm 26

27 Page 27 CCM6+7+ UNIT 9 GEOMETRY 2D and 3D 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. 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

28 Page 28 CCM6+7+ UNIT 9 GEOMETRY 2D and 3D 28

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30 Page 30 CCM6+7+ UNIT 9 GEOMETRY 2D and 3D On the graph below, graph point A(-2, 3) as the center of a circle with a radius of 4. Find the area and circumference of the circle

31 Page 31 CCM6+7+ UNIT 9 GEOMETRY 2D and 3D 31

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33 Page 33 CCM6+7+ UNIT 9 GEOMETRY 2D and 3D 3-Dimensional Figures and Nets Practice Name each 3-D figure and count the number of faces. Then, draw the nets for each figure

34 Page 34 CCM6+7+ UNIT 9 GEOMETRY 2D and 3D Match each cross section with the appropriate 3-D figure Write the definition of the following: 8.) Net: 9.) 3-Dimensional Figure: 10.) Cross-Section: 34

35 Page 35 CCM6+7+ UNIT 9 GEOMETRY 2D and 3D CROSS SECTIONS Name: Date: Period: 352. Below are three cross sections of a pyramid with a square base. Color or shade in the cross sections and then name the shape of the cross section. SHAPE OF THE PYRAMID BASE: SHAPE OF ANY CROSS SECTION THAT IS CUT PARALLEL TO THE BASE: 352. Below each figure, draw the shape that you think would be created if the dotted line represents a cut that a plane made through the figure Below each cylinder, draw the shape that you think would be created if the dotted line represents a plane that cuts through the figure. 35

36 Page 36 CCM6+7+ UNIT 9 GEOMETRY 2D and 3D 352. Below each cone, draw the shape that you think would be created if the dotted line represents a plane that cuts through the figure. Complete the following statement: When slicing a cross section that is parallel to the base, the cross section shape will always. 36

37 Page 37 CCM6+7+ UNIT 9 GEOMETRY 2D and 3D CROSS SECTIONS GUIDED PRACTICE A G B F K Z P H J M C E I D Using the cube above, describe what the shape of the cross-section would look like if the plane that cuts it goes through the given points. 1. Through points : J, M, H 2. Through points: P, K, M 3. Through points: E, C, K, P 4. Through points: C, D, A, F 5. Through points: M, H, E 6. Through points: G, K, M, H 7. No matter what the original 3D figure, if you cut with a plane parallel to the base your cross section will always be. 37

38 Page 38 CCM6+7+ UNIT 9 GEOMETRY 2D and 3D SURFACE AREA is. A Student Friendly Model of Finding Surface Area of a Rectangular Prism Once students have completed Exploring 3-D Rectangular Prisms, it may be helpful to introduce a way to find surface area of rectangular prisms that may help visual learners. If these were the dimensions of a rectangular prism, have students make a smiley face 4 x 12 x 8 Each connector can be used to find the surface area of the rectangular prism: 4 x x 8 4 x 8 Since students should have established that there will be two identical faces for each set of dimensions, they can set up finding the surface area as follows: 4 x 12 = 48 x 2 = x 8 = 96 x 2 = x 8 = 32 x 2 = square units 38

39 Page 39 CCM6+7+ UNIT 9 GEOMETRY 2D and 3D 39

40 Page 40 CCM6+7+ UNIT 9 GEOMETRY 2D and 3D 40

41 Page 41 CCM6+7+ UNIT 9 GEOMETRY 2D and 3D 41

42 Page 42 CCM6+7+ UNIT 9 GEOMETRY 2D and 3D Wrapping Up Surface Area Find the surface area of each three-dimensional figure. Show all steps and work to support your answers. 42

43 Page 43 CCM6+7+ UNIT 9 GEOMETRY 2D and 3D 43

44 Page 44 CCM6+7+ UNIT 9 GEOMETRY 2D and 3D If the length, width, and height of a rectangular prism are 4 yards, 8 yards, and 11 yards respectively, what would be the surface area? 44

45 Page 45 CCM6+7+ UNIT 9 GEOMETRY 2D and 3D How Do YOU think you would find the Surface area of a cylinder? 1) Draw a net. Let s say the radius of the circle is 10 cm and the height of the cylinder is 20 cm. 2) Find the area of the circles. 3) The rest of the net is a big rectangle what are the dimensions? L = W = What is the area of the rectangle? TOTAL SURFACE AREA = cm 2 FORMULA FOR SA of a cylinder: 45

46 Page 46 CCM6+7+ UNIT 9 GEOMETRY 2D and 3D 46

47 Page 47 CCM6+7+ UNIT 9 GEOMETRY 2D and 3D For ALL SHAPES: V = Bh **or very close to it where B is the area of the base shape and h is the height of the 3-D figure 47

48 Page 48 CCM6+7+ UNIT 9 GEOMETRY 2D and 3D Volume of Cylinders: Volume of Pointy Shapes: Cones: Pyramids: So Pointy Shapes are V = Bh Volume of Spheres: Volume of a Sphere = 48

49 Find the Volume of each shape below. Page 49 CCM6+7+ UNIT 9 GEOMETRY 2D and 3D 49

50 Page 50 CCM6+7+ UNIT 9 GEOMETRY 2D and 3D 50

51 FRACTIONAL EDGE LENGTHS Problem Page 51 CCM6+7+ UNIT 9 GEOMETRY 2D and 3D 1 1 A right rectangular prism has edges of 1, 1 and 1. How many cubes with side lengths of prism? What is the volume of the prism? would be needed to fill the 2 Fractional Volume Word Problems Part I 1. A right rectangular prism has edges of, in, 2 in and in. How many cubes with lengths of 1 4 in would be needed to fill the prism? What is the volume? 2. Find the volume of a rectangular prism with dimensions 1 1 in, 1 1 in and 2 1 in.how many cubes with lengths of 1 2 in would be needed to fill the prism? 3. A follower box is 3feet long, 1 3 feet wide, and 1 feet deep. How many cubic feet of dirt can it 4 2 hold? 51

52 Page 52 CCM6+7+ UNIT 9 GEOMETRY 2D and 3D Fractional Volume Word Problems Part II 4. Draw a diagram to match the rectangular prism whose length is 5 1 in, width is 4in and height is in. 5. Use centimeter grid papers to build a rectangular prism with the volume of 24 cubic units. At least one of the side lengths of the prism is a fractional unit. What are the dimensions of the rectangular prisms you built? What is the surface area of the prism? 6. Mr. White is trying to store boxes in a storage room with length of 8yd, width of 5yd and height of 2yd. How many boxes can fit in this space if each is box is feet long 11 2 feet wide and 1 feet deep? 52

53 Page 53 CCM6+7+ UNIT 9 GEOMETRY 2D and 3D 53

54 Page 54 CCM6+7+ UNIT 9 GEOMETRY 2D and 3D 54

55 Page 55 CCM6+7+ UNIT 9 GEOMETRY 2D and 3D UNIT 9 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 area of the rectangle if the perimeter of this rectangle is

56 WORD PROBLEMS and FORMULAS Page 56 CCM6+7+ UNIT 9 GEOMETRY 2D and 3D 14) The perimeter of a rectangle is 12. Determine a possible length and width, then calculate a possible area for that rectangle. 15) 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? 16) 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) meters 17) 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 18) 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 19) The perimeter of a rectangle is 20 ft 2. If the length is 5 ft, what is the AREA of the rectangle? 56

57 Page 57 CCM6+7+ UNIT 9 GEOMETRY 2D and 3D 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. 20. G (-4, 5) H (5, 5) I (-4, -5) J (5, -5) Perimeter of GHIJ: Area of GHIJ: 57

58 Page 58 CCM6+7+ UNIT 9 GEOMETRY 2D and 3D 21. 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

59 Page 59 CCM6+7+ UNIT 9 GEOMETRY 2D and 3D Find the Area and Perimeter. 59

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