# Chapter 4: Area, Perimeter, and Volume. Geometry Assessments

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1 Chapter 4: Area, Perimeter, and Volume Geometry Assessments

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3 Area, Perimeter, and Volume Introduction The performance tasks in this chapter focus on applying the properties of triangles and polygons to compute area, perimeter, and volume. Chapter 4: Area, Perimeter, and Volume 35

4 Area, Perimeter, and Volume 36 Chapter 4: Area, Perimeter, and Volume

5 Area, Perimeter, and Volume Boxing Basketballs 1. A basketball (sphere) with a circumference of approximately 30 inches is packed in a box (cube) so that it touches each side of the interior of the box. Answer the following question, ignoring the thickness of the surface of the ball and the surface of the box. What is the volume of the wasted space in the box?. A box (cube) that has a side length of 9.5 inches is packed inside a ball (sphere) so that the corners of the box touch the interior of the sphere. Answer the following question, ignoring the thickness of the surface of the ball and the surface of the box. What is the volume of the wasted space in the ball? Chapter 4: Area, Perimeter, and Volume 37

6 Area, Perimeter, and Volume Materials: Notes One pencil and graphing calculator per student Geometry TEKS Focus: (G.5) Geometric patterns. The student uses a variety of representations to describe geometric relationships and solve problems. The student is expected to: (D) identify and apply patterns from right triangles to solve meaningful problems, including special right triangles ( and ) and triangles whose sides are Pythagorean triples. (G.8) Congruence and the geometry of size. The student uses tools to determine measurements of geometric figures and extends measurement concepts to find perimeter, area, and volume in problem situations. The student is expected to: (D) find surface areas and volumes of prisms, pyramids, spheres, cones, cylinders, and composites of these figures in problem situations. Additional Geometry TEKS: (G.8) Congruence and the geometry of size. The student uses tools to determine measurements of geometric figures and extends measurement concepts to find area, perimeter, and volume in problem situations. The student is expected to: (C) derive, extend, and use the Pythagorean Theorem; and Teacher Notes Scaffolding Questions: For the first situation, how do the cube and the sphere come in contact with each other (sides or vertices)? Do the cube and the sphere have any common measurements? In the second problem, how do the cube and the sphere come in contact with each other? What triangle is formed by the diagonal of the base of the cube and the edges of the base of the cube? What triangle is formed by the diagonal of the cube, the diagonal of the base, and one vertical edge of the cube? Sample Solutions: 1. First, find the volumes of the ball and the box. The diameter of the ball is the side length of the box. Use the circumference, C, to find the diameter, d. C = d 30 = d d 9.55 in The volume of the box, V, in terms of the length of the side, s, is found by using the formula V = s 3. V 955. V in 3 V V Find the volume of the ball. r 955. V in Chapter 4: Area, Perimeter, and Volume

7 Area, Perimeter, and Volume Subtract the volume of the ball from the volume of the box to find the volume of wasted space in in 3 = in 3. If the cube is surrounded by the ball, the cube will touch the interior of the sphere at eight vertex points. To determine the diameter of the sphere we must find the diagonal of the cube. Since the diagonal of the cube passes through the center of the sphere and its endpoints touch the sphere, its length is the diameter of the sphere. Connections to TAKS: Objective 6: The student will demonstrate an understanding of geometric relationships and spatial reasoning. Objective 8: The student will demonstrate an understanding of the concepts and uses of This can be found by using the Pythagorean Theorem or special right triangles. Chapter 4: Area, Perimeter, and Volume 39

8 Area, Perimeter, and Volume First find a diagonal of a side of the cube. The right triangle has sides measuring 9.5 inches. 9.5 c c c Now use the Pythagorean Theorem again to find the diagonal of the cube. This diagonal is the hypotenuse of a right triangle. One leg of the triangle is the side of the square, which measures 9.5 inches, and the other leg is the diagonal of the side face, 95. inches d inches d Now find the volume of the sphere and the cube. Cube V s 3 V V in in Sphere V V r V in The volume of the wasted space is the volume of the sphere minus the volume of the cube = The volume of the wasted space is approximately in Chapter 4: Area, Perimeter, and Volume

9 Area, Perimeter, and Volume Extension Questions: If the circumference of the ball were doubled in problem 1, how would the wasted space be affected? If the circumference is doubled, the diameter and the radius would also be doubled. The volume should be multiplied by 3 or 8. This idea may be demonstrated by looking at the general formula. The volume of the wasted space, in terms of the diameter for the original situation, is d d d If the diameter is doubled, the equation becomes 3 4 d d d d 3 The wasted space is multiplied by 8. If the length of the side of the box in problem is multiplied by one-third, how will the diameter of the sphere be affected? The diameter will also be multiplied by one-third. The diameter of the sphere is the diagonal of the cube. The diagonal of the cube is the hypotenuse of a right triangle. The right triangle is similar to the right triangle in the original cube. The corresponding sides of the two right triangles are proportional. Since the side of the new cube (also the leg of the right triangle) is one-third times the side of the original cube, the diagonal of the new cube (also the hypotenuse of the similar triangle ) is one-third times the diagonal of the original cube. Chapter 4: Area, Perimeter, and Volume 41

10 Area, Perimeter, and Volume Student Work Sample The work on the next page was performed as an individual assessment following a unit on volumes. The student s work exemplifies many of the criteria from the solution guide. Note the following criteria: Shows an understanding of the relationships among elements. The student uses arrows in the first problem to show that circumference is used to determine the diameter, which is equal to the length of the box s edges. In the second problem, he shows how he used the edge of the box and the right triangles to find the radius of the sphere. He shows how he determines the answers in both problems by subtracting the two volumes, the volume of the box, and the volume of the sphere. States a clear and accurate solution using correct units. The student not only uses the correct units in 3 but indicates that it is the volume of the wasted space. 4 Chapter 4: Area, Perimeter, and Volume

11 Area, Perimeter, and Volume Chapter 4: Area, Perimeter, and Volume 43

12 Area, Perimeter, and Volume 44 Chapter 4: Area, Perimeter, and Volume

13 Area, Perimeter, and Volume Flower A landscape company has produced the garden design below (see figure 1). The petals are constructed by drawing circles from the vertex to the center of a regular hexagon with a radius of 0 feet (see figure ). figure 1 figure The company plans to plant flowers inside the area of each petal of the garden design. The company must know the area of the petals in order to determine how many flowers to purchase for planting. Find the area of the petals. Give answers correct to the nearest hundredth. Chapter 4: Area, Perimeter, and Volume 45

14 Area, Perimeter, and Volume measurement and similarity. Materials: Notes One graphing calculator per student Geometry TEKS Focus: (G.8) Congruence and the geometry of size. The student uses tools to determine measurements of geometric figures and extends measurement concepts to find perimeter, area, and volume in problem situations. The student is expected to: (A) find areas of regular polygons, circles, and composite figures; (B) find areas of sectors and arc lengths of circles using proportional reasoning; Additional Geometry TEKS: (G.5) Geometric patterns. The student uses a variety of representations to describe geometric relationships and solve problems. The student is expected to: (D) identify and apply patterns from right triangles to solve meaningful problems, including special right triangles ( and ) and triangles whose sides are Pythagorean triples. Teacher Notes Scaffolding Questions: Are all of the petals the same size? What is the combination of figures that form a petal? How can the measure of the arc be determined? Sample Solutions: The hexagon eases the solution in that it can be subdivided into six equilateral triangles. First, find the area of one of the circles. A r A ft Each angle of the hexagon is 10º. Find the area of the sector of the circle. 10 The hexagon is composed of six equilateral triangles. The area of half the sector shown is 60 of the area of the circle. Thus, to determine the area of the sector, multiply the area of the circle by one-sixth or ft A 46 Chapter 4: Area, Perimeter, and Volume

15 Area, Perimeter, and Volume In order to find the area of the segments of the circle that form a petal, the area of the triangle must be subtracted from the area of the one-sixth sector of the circle. Determine the height of the triangle. h h h ft height 10 0 (G.8) Congruence and the geometry of size. The student uses tools to determine measurements of geometric figures and extends measurement concepts to find perimeter, area, and volume in problem situations. The student is expected to: (C) derive, extend, and use the Pythagorean Theorem; and Connections to TAKS: Objective 8: The student will demonstrate an understanding of the concepts and uses of measurement and similarity. The area of the triangle is 1 1 A bh ( 0)( 17. 3) 173. ft The area of one-half of a petal is the area of the sector minus the area of the triangle = 36.4 ft The area of one petal is (36.4) ft = 7.48 ft Multiplying by 6 yields the total area of the petals. Total area = 6(7.48) = ft Chapter 4: Area, Perimeter, and Volume 47

16 Area, Perimeter, and Volume Extension Questions: If the length of the hexagon side is doubled, how is the area affected? If the length is doubled, then the area is multiplied by a factor of two squared. Is it possible to use other regular polygons to create flowers using the same process? What makes this activity possible is the fact that the side of the hexagon becomes the radius of the circle. The hexagon is the only one for which this relationship is true. 48 Chapter 4: Area, Perimeter, and Volume

17 Area, Perimeter, and Volume Chapter 4: Area, Perimeter, and Volume 49

18 Area, Perimeter, and Volume Student Work Sample The next page shows the work of an individual student. The teacher required her students to give a verbal description of the process used to solve the problem. The second page shows the student s computations. The work exemplifies the following criteria: Communicates clear, detailed, and organized solution strategy. The student provides a step-by-step description of the steps. He gives a reason for each step. Demonstrated geometric concepts, process, and skills. She used the Pythagorean Theorem. She showed how to get the area of the hexagon and the area of a circle. She showed how to find the area of the petal region by subtracting the area of the hexagon from the area of the circle. 50 Chapter 4: Area, Perimeter, and Volume

19 Area, Perimeter, and Volume Chapter 4: Area, Perimeter, and Volume 51

20 Area, Perimeter, and Volume 5 Chapter 4: Area, Perimeter, and Volume

21 Area, Perimeter, and Volume Great Pyramid King Khufu, the second pharaoh of the fourth Egyptian dynasty, commissioned the building of the Great Pyramid of Giza. This monument was built of solid stone and was completed in just under 30 years, around 560 BCE. It presides over the plateau of Giza on the outskirts of Cairo, and is the last survivor of the Seven Wonders of the World. At 481 feet high, the Great Pyramid stood as the tallest structure in the world for more than 4,000 years. The base of the Great Pyramid was a square with each side measuring 756 feet. 1. If the average stone used in the construction was a cube measuring 3.4 feet on a side, approximately how many stones were used to build this solid monument? Justify your answer.. If you could unfold the pyramid like the figure below, how many acres of land would it cover? (1 acre = 43,560 square feet) Chapter 4: Area, Perimeter, and Volume 53

22 Area, Perimeter, and Volume Materials: Notes One graphing calculator per student Geometry TEKS Focus: (G.6) Dimensionality and the geometry of location. The student analyzes the relationship between three-dimensional geometric figures and related two-dimensional representations and uses these representations to solve problems. The student is expected to: (B) use nets to represent and construct three-dimensional geometric figures; and (G.8) Congruence and the geometry of size. The student uses tools to determine measurements of geometric figures and extends measurement concepts to find perimeter, area, and volume in problem situations. The student is expected to: (D) find surface areas and volumes of prisms, pyramids, spheres, cones, cylinders, and composites of these figures in problem situations. Teacher Notes Scaffolding Questions: What shape is the solid figure? How do you know? What formula do you need in order to find the number of stones? Explain what measurements you will use in order to determine the volume of the pyramid. What are the component parts of the net of the pyramid? Describe how you could find the surface area of the pyramid. Sample Solutions: 1. Find the volume of a stone and the pyramid. Stone V s 3 V V ft 3 Pyramid 1 V Bh 3 1 V Vpyramid Number of stones, 331, V stone V 91, 636, 7 ft If whole number of stones are required, the number of stones would be,33, Find the area of component parts. Square A s A 756 A ft Triangle Find the slant height of the pyramid; this is the hypotenuse of the cross section of the pyramid. a b c slant height slant height ft slant height 54 Chapter 4: Area, Perimeter, and Volume

23 Area, Perimeter, and Volume Now find the area of a side. Extension Questions: If all the dimensions of the pyramid were multiplied by 4, how would the volume of the pyramid be changed? 1 A bh 1 A A ft The four sides of the pyramid are congruent. The total surface area of the pyramid is the area of the square base plus the area of the four sides. Total area Convert to acres: ( ) ft ft 1 acre ft 34 acres If a side is multiplied by a factor of 4, the volume is multiplied by a factor of 4 3. If you want to have the net cover twice the surface area, how would the sides of the pyramid need to be changed? If a side is multiplied by a factor of k, the area is multiplied by a factor of k. If the area is multiplied by a factor of h, the side is multiplied by a factor of h. Additional Geometry TEKS: (G.8) Congruence and the geometry of size. The student uses tools to determine measurements of geometric figures and extends measurement concepts to find perimeter, area, and volume in problem situations. The student is expected to: (C) derive, extend, and use the Pythagorean Theorem; and (G.9) Congruence and the geometry of size. The student analyzes properties and describes relationships in geometric figures. The student is expected to: (D) analyze the characteristics of polyhedra and other three-dimensional figures and their component parts based on explorations and concrete models. Connections to TAKS: Objective 7: The student will demonstrate an understanding of two- and three-dimensional representations of geometric relationships and shapes. Objective 8: The student will demonstrate an understanding of the concepts and uses of measurement and similarity. Chapter 4: Area, Perimeter, and Volume 55

24 Area, Perimeter, and Volume 56 Chapter 4: Area, Perimeter, and Volume

25 Area, Perimeter, and Volume Walter and Juanita s Water Troughs You have been hired as chief mathematician by a company named Walter and Juanita s Water Troughs. This company builds water troughs for various agricultural uses. The company has one design (see figure 1.). Your job is to perform mathematical analysis for the owners. figure 1 ft ft ft 10 ft 1. A customer would like to know what the depth of the water is (in inches) if the trough has only 3 gallons in it.. The interior of the troughs must be coated with a sealant in order to hold water. One container of sealant covers 400 square feet. Will one container of sealant be enough to seal ten troughs? Why or why not? 3. Walter and Juanita would like to explore some minor modifications of their original design. They would like to know which change will produce a water trough that would hold more water adding one foot to the length of the trough, making it 11 feet long, or adding three inches to each side of the triangular bases, making them feet 3 inches on each side (see figures and 3). Justify your answer. figure figure 3 ft ft 3 in ft ft 11 ft ft 3 in ft 3 in 10 ft Chapter 4: Area, Perimeter, and Volume 57

26 Area, Perimeter, and Volume Materials: Notes One calculator and straightedge per student Geometry TEKS Focus: (G.5) Geometric patterns. The student uses a variety of representations to describe geometric relationships and solve problems. The student is expected to: (D) identify and apply patterns from right triangles to solve meaningful problems, including special right triangles ( and ) and triangles whose sides are Pythagorean triples. Teacher Notes Scaffolding Questions: What important components of the trough are needed to solve the problems? What is your prediction for which trough will hold the most water in number 3? Justify your answer. Sample Solutions: 1. To solve this problem, the area of the triangular base must be found in terms of an unknown side. If water is poured into the trough, the height of the water is a function of the side of the triangle. Consider the end of the trough that is an equilateral triangle. (G.8) Congruence and the geometry of size. The student uses tools to determine measurements of geometric figures and extends measurement concepts to find perimeter, area, and volume in problem situations. 30 x x The student is expected to: (D) find surface areas and volumes of prisms, pyramids, spheres, cones, cylinders, and composites of these figures in problem situations. Using special right triangle properties or the x 3 Pythagorean Theorem, the altitude is. ft Area of the base can be found by: Area 1 x 3 x x 3 Area ft 4 The desired volume is given in gallons and must be converted to cubic feet because the measurements are given in feet. 3 gal. 3 1 ft 48. ft 748. gal 3 The volume of the tank is the area of the triangular base times the length of the tank. 58 Chapter 4: Area, Perimeter, and Volume

27 Area, Perimeter, and Volume x ft x x ft The side length is ft. 3 The depth of the water is the altitude of the base altitude altitude ft depthof water in. Find the interior surface area of one trough and multiply by 10. Figure : First, find the area of the base (the ends of the trough). In order to do this, find the altitude of the triangle. The base is an equilateral triangle, so the altitude bisects the side and the intercepted angle. The angles of an equilateral triangle are all 60º. Additional Geometry TEKS: (G.8) Congruence and the geometry of size. The student uses tools to determine measurements of geometric figures and extends measurement concepts to find perimeter, area, and volume in problem situations. The student is expected to: (A) find areas of regular polygons, circles, and composite figures; (C) derive, extend, and use the Pythagorean Theorem; and Connections to TAKS: Objective 8: The student will demonstrate an understanding of the concepts and uses of measurement and similarity. ft 3030 ft 1ft ft 1ft Use the Pythagorean Theorem to find the altitude. a b c a a a 3 a 3 Chapter 4: Area, Perimeter, and Volume 59

28 Area, Perimeter, and Volume The altitude must be positive. a 3 Another approach to determining the altitude is to use special right triangle properties. The hypotenuse is twice the shortest side. The side opposite the 60-degree angle is the shorter leg times ft 30 ft The altitude is ft ft The area of the triangular base may be found using the altitude and the base of the triangle. 1 1 Area ft bh The surface area is the sum of the areas of the ends and the sides ft The surface are of ten troughs is 10 times the surface area of one trough ft Since the gallon of paint covers 400 ft, there will not be enough sealant to seal ten troughs. 3. The volume of a right prism is the area of the base times the height. The area of the base for the first case was found in problem. 1 1 Area bh ft V Bh ( 11) ft 3 60 Chapter 4: Area, Perimeter, and Volume

29 Area, Perimeter, and Volume For the second case, find the altitude of the right triangle base with side of.5 feet and the shorter leg, one-half of.5 or Use the Pythagorean Theorem. a b c a a a a 195. ft Find the area of the base. 1 A bh 1 A A 19. ft Find the volume. V Bh V ( 19. )( 10) V 1. 9 ft 3 Adding 3 inches to each side of the base will produce a greater increase in volume than adding a foot to the distance between the bases. Chapter 4: Area, Perimeter, and Volume 61

30 Area, Perimeter, and Volume Extension Question: A new tank is designed in the shape of a hemisphere (half of a sphere) to hold the same volume of water as the tank in figure 1. What is the depth of the water if the tank is full? The area of the base (the ends of the trough) is the same as the area for figure. The area is ft 3. The volume of a right prism is the area of the base times the height. V=Bh ft Volume of a sphere = r, but we want only of this amount. 3 1 V r r r r r r r 3 The depth is approximately.0 feet. 6 Chapter 4: Area, Perimeter, and Volume

31 Area, Perimeter, and Volume Greenhouse The backyard greenhouse in the figure below uses plastic tubing for framing and plastic sheeting for wall covering. The end walls are semicircles, and the greenhouse is built to the dimensions in the figure below. All walls and the floor are covered by the plastic sheeting. The door is formed by cutting a slit in one of the end walls. 10 ft 10 ft 1. How many square feet of plastic sheeting will it take to cover the top, sides, and floor of the greenhouse?. What is the volume of the greenhouse? Chapter 4: Area, Perimeter, and Volume 63

32 Area, Perimeter, and Volume Materials: Notes One graphing calculator per student Geometry TEKS Focus: (G.8) Congruence and the geometry of size. The student uses tools to determine measurements of geometric figures and extends measurement concepts to find perimeter, area, and volume in problem situations. The student is expected to: (A) find areas of regular polygons, circles, and composite figures; (D) find surface areas and volumes of prisms, pyramids, spheres, cones, cylinders, and composites of these figures in problem situations. Teacher Notes Scaffolding Questions: Describe the shape of the greenhouse. What dimensions do you need to determine the surface area? Sample Solutions: 1. The ends of the greenhouse are semicircles that have a diameter of ten. Area of each end: A A r The area of one-half of the circle is A ft Area of side: Since the greenhouse is half of a cylinder, we can unwrap the sides, find the area of the rectangle, and divide by. One dimension of the rectangle is the circumference, and the other is 10. C d C 10 C ft Find the area. Surface area of the cylinder ft Surface area of half of the cylinder 314 ft 157 ft 1 The floor is the area of the square. A A 100 ft 64 Chapter 4: Area, Perimeter, and Volume

33 Area, Perimeter, and Volume Total square feet of plastic needed is = ft.. The volume of the greenhouse is the area of the base multiplied by the distance between the bases. The base is a semicircle with a diameter of A r because we only need half of the circle. 1 A 5 A ft The volume of the greenhouse may be computed using the formula. V Bh V V ft 3 Additional Geometry TEKS: (G.8) Congruence and the geometry of size. The student uses tools to determine measurements of geometric figures and extends measurement concepts to find perimeter, area, and volume in problem situations. The student is expected to: (B) find areas of sectors and arc lengths of circles using proportional reasoning; Connections to TAKS: Objective 8: The student will demonstrate an understanding of the concepts and uses of measurement and similarity. Extension Question: What would the dimensions of a new square floor greenhouse have to be in order to double the volume of the greenhouse in this problem? Let the side length be x. The radius of the semicircular ends would be expressed as The area of the base is one-half of the area of a circle with radius 1 1 A x A 1 8 x x. The volume of the greenhouse is the area of the base (the area of the semicircle) times the length, x. x. Chapter 4: Area, Perimeter, and Volume 65

34 Area, Perimeter, and Volume The rule for the volume as a function of the side is: 1 V x x 8 The volume must two double the original volume or two times 39.7 ft 3. The volume must be ft x 8 3 x 1. 6 ft The length of the side of the base must be about 1.6 feet. 66 Chapter 4: Area, Perimeter, and Volume

35 Area, Perimeter, and Volume Nesting Hexagons The first four stages of nested hexagons are shown below. Connecting the midpoints of the sides of the previous stage creates each successive stage. The side of the hexagon in Stage 0 measures 3 units. Stage 1 Stage 1 Stage Stage 3 1. Create a function rule to find the area of the innermost hexagon of any stage.. Find the area of the innermost hexagon in stage What is the domain of your function rule? 4. What is the range of your function rule? Chapter 4: Area, Perimeter, and Volume 67

36 Area, Perimeter, and Volume Materials: Notes One graphing calculator and straightedge per student Geometry TEKS Focus: (G.5) Geometric patterns. The student uses a variety of representations to describe geometric relationships and solve problems. The student is expected to: (A) use numeric and geometric patterns to develop algebraic expressions representing geometric properties; (G.8) Congruence and the geometry of size. The student uses tools to determine measurements of geometric figures and extends measurement concepts to find perimeter, area, and volume in problem situations. The student is expected to: (A) find areas of regular polygons, circles, and composite figures; Additional Geometry TEKS: (G.5) Geometric patterns. The student uses a variety of representations to describe geometric relationships and solve problems. The student is expected to: (B) use numeric and geometric patterns to make generalizations about geometric properties, including properties of polygons, ratios in similar figures and solids, and angle relationships in polygons and circles; Teacher Notes Note: It may prove beneficial to create a geometry software sketch of Nesting Hexagons once students have performed the calculations for Stages 1 and. Students can explore whether the size of the hexagon has any impact on the relationship between the stages. They can also use the measurements to help determine a function rule for this situation. It may also prove beneficial to model the organization of data in a table such as the one in the sample solutions. Encourage students to round area values to the nearest hundredth. Scaffolding Questions: If students have created a table, the following questions may be asked: Is there a common difference between the terms in the area column of the table? Is there a common second difference between the terms in the table? Is there a common ratio between the terms in the table? How is each stage related to the previous stage? Sample Solutions: 1. The perimeter is 18. Use A 1 ap to find the area of the regular hexagon. Each interior angle of a regular hexagon is 10. The segments from each vertex to the center bisect the interior angles forming six equilateral triangles. Therefore the radius and the sides of the hexagon have the same measure. Using right triangle properties, the length of the apothem is found by dividing the hypotenuese, or in this case the radius, by and multiplying by 3. The length of the apothem for stage 0 is 3 3, and the length of the. radius is 3. The perimeter, p, is 6 times 3 units, or 18 units. 68 Chapter 4: Area, Perimeter, and Volume

37 Area, Perimeter, and Volume (D) identify and apply patterns from right triangles to solve meaningful problems, including special right triangles ( and ) and triangles whose sides are Pythagorean triples. Connections to TAKS: Objective 1: The student will describe functional relationships in a variety of ways. 1 A ap 1 A A A Stage Apothem Radius Side Length Perimeter Area The apothem of the stage 0 hexagon is now the radius of the stage 1 hexagon. The length of the side of the stage 1 hexagon is square units Area rounded to the nearest hundredth Objective 6: The student will demonstrate an understanding of geometric relationships and spatial reasoning. Objective 8: The student will demonstrate an understanding of the concepts and uses of measurement and similarity. Note: It may prove beneficial to create a geometry software sketch of Nesting Hexagons once students have performed the calculations for Stages 1 and. Students can explore whether the size of the hexagon has any impact on the relationship between the stages. They can also use the measurements to help determine a function rule for this situation. It may also prove beneficial to model the organization of data in a table such as the one in the possible solution strategies. Encourage students to round area values to the nearest hundredth. Using right triangle properties, the length of the side is, and the length of the apothem is Chapter 4: Area, Perimeter, and Volume 69

38 Area, Perimeter, and Volume 1 A ap 1 A A A 7 3 or A square units Stage Apothem Radius Side Length Perimeter Area Area rounded to the nearest hundredth A pattern emerges when the rest of the table is completed. In each new stage the side length, radius, apothem, and perimeter change by a factor of previous stage. The area in the new stage changes by a factor of the previous stage. 3 times the amount in the 3 4 times the amount in 70 Chapter 4: Area, Perimeter, and Volume

39 Area, Perimeter, and Volume Stage Apothem Radius Side Length Perimeter Area Area rounded to the nearest hundredth A recursive process of repeated multiplication can be used to generate the function rule for the area. Stage Area Another approach is to look for patterns in the area values themselves. It will facilitate the development of a pattern if the values are rounded to the nearest hundredth. If the difference between each term is found, no pattern emerges. If the difference between each of those differences is found, no pattern emerges. However, if the ratios of the terms are found, a common ratio emerges. This means that each successive term can be found by multiplying the previous term by this common ratio. This leads to the generation of the exponential rule for this situation. Chapter 4: Area, Perimeter, and Volume 71

40 Area, Perimeter, and Volume A 7 3 x A Possible notation for the domain of the function might include one of the following: all real numbers, x, or -,+. The domain for the problem situation would be non-negative integers. 4. Possible notation for the range of the function might include one of the following: real numbers greater than 0, 0 y, or 0,+. The range for the problem situation is more complicated. The range is the specific set of numbers corresponding to the domain values for the problem situation. For example, , 8, 3, Chapter 4: Area, Perimeter, and Volume

41 Area, Perimeter, and Volume Extension Questions: stagen What is the relationship between the ratio of side lengths stagen 1 and the ratio of perimeters? How is this number related to the ratio of areas? The ratio of side lengths and the ratios of perimeters are both is the ratio of the areas. 3.. This ratio squared What is the largest possible area in this problem? Justify your answer. The largest possible area is or The area of each successive stage is 3 4 times the area of the hexagon of the previous stage, so the area is always smaller than the area of the hexagon of stage 0. What is the smallest possible area in this problem? Justify your answer. There will never be a smallest area. Any area thought to be the smallest area can be multiplied by according to our rule in order to find a new stage with a smaller area. The area will approach 0 as the stage number gets very large but will never actually be Chapter 4: Area, Perimeter, and Volume 73

42 Area, Perimeter, and Volume 74 Chapter 4: Area, Perimeter, and Volume

43 Chapter 5: Solids and Nets Geometry Assessments

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45 Solids and Nets Introduction The performance tasks in this chapter require students to analyze relationships between two- and three-dimensional objects and use these relationships to solve problems. Chapter 5: Solids and Nets 77

46 Solids and Nets 78 Chapter 5: Solids and Nets

47 Solids and Nets Perfume Packaging Gina would like to package her newest fragrance, Persuasive, in an eyecatching yet cost-efficient box. The Persuasive perfume bottle is in the shape of a regular hexagonal prism 10 centimeters high. Each base edge of the prism is 3 centimeters. The cap is a sphere with a radius of 1.5 centimeters. Gina would like to compare the cost of the proposed box designs shown below. It is important that each bottle fit tightly in the box to avoid movement during shipping, but the box must be large enough to allow people to get the bottle in and out of it. Thus, the box should only be 0.5 cm taller and 0.5 cm wider than the height and widest portion of the base of the bottle. 1. Determine the measurements needed for each dimension on the two boxes. Explain how you determined these dimensions.. Sketch the net for each package design, showing how you would make each box from a single sheet of cardboard. (Do not include the flaps needed to glue the box.) 3. Calculate the amount of cardboard used for each design. Which box will be more cost-efficient? Chapter 5: Solids and Nets 79

48 Solids and Nets Materials: Notes One graphing calculator per student Geometry TEKS Focus: (G.6) Dimensionality and the geometry of location. The student analyzes the relationship between three-dimensional geometric figures and related two-dimensional representations and uses these representations to solve problems. The student is expected to: (B) use nets to represent and construct three-dimensional geometric figures; and (G.8) Congruence and the geometry of size. The student uses tools to determine measurements of geometric figures and extends measurement concepts to find perimeter, area, and volume in problem situations. The student is expected to: (A) find areas of regular polygons, circles, and composite figures; (D) find surface areas and volumes of prisms, pyramids, spheres, cones, cylinders, and composites of these figures in problem situations. Additional Geometry TEKS: (G.5) Geometric patterns. The student uses a variety of representations to describe geometric relationships and solve problems. The student is expected to: (D) identify and apply patterns from right triangles to solve Teacher Notes Scaffolding Questions: What is the total height of the perfume bottle and the cap? Describe the shape of the sides of each box. Describe the bases of each box. How will the dimensions of the perfume bottle compare to the dimensions of the box? Sample Solutions: 1. The total height of the perfume bottle and cap is 13 centimeters. The base of the bottle is a regular hexagon with sides of 3 centimeters. A sketch of the base is shown below. The diagonals of the regular hexagon intersect in the center of the hexagon. The segment connecting the center of the regular hexagon and a vertex is the radius, AP or BP, of the regular hexagon. A 3 cm P B N C A regular hexagon has a radius equal to the length of the side of the regular hexagon. This can be established by drawing in the apothem of the hexagon and using a triangle. The regular hexagon has an interior angle, ABC, measuring 10 degrees. The radius of the hexagon, BP, bisects this angle forming a 60-degree angle, CBP. The apothem PN forms a 90-degree angle with the side of the hexagon. The apothem also bisects the side of the hexagon; therefore, 1.5 centimeters. The remaining angle of the triangle, measures 30 degrees. BN measures BNP, 80 Chapter 5: Solids and Nets

49 Solids and Nets Using this information, it can be determined that the hypotenuse of the triangle is 3 centimeters. The hypotenuse of the triangle is also the radius of the hexagon. The following figure shows the measurements in the triangle. Measurements are rounded to the nearest hundredth. 3 cm cm The required 0.5 cm allowance for packaging space must be considered. This changes the length of the diagonal of the hexagon to 6.5 cm. The diagonal is made up of radii (shown as the hypotenuse in the triangle below). Therefore the actual measurement needed for the packaging must be adjusted on the triangle as follows: cm meaningful problems, including special right triangles ( and ) and triangles whose sides are Pythagorean triples. (G.8) Congruence and the geometry of size. The student uses tools to determine measurements of geometric figures and extends measurement concepts to find perimeter, area, and volume in problem situations. The student is expected to: (B) find areas of sectors and arc lengths of circles using proportional reasoning; Connections to TAKS: Objective 7: The student will demonstrate an understanding of two-and three-dimensional representations of geometric relationships and shapes cm.81 cm Objective 8: The student will demonstrate an understanding of the concepts and uses of measurement and similarity cm The widest dimension of the base will be 6.5 centimeters. This dimension will be the same for the hexagon and the circular base. The shorter distance across the hexagon ( apothems, end to end) will be approximately 5.6 cm. (This amount is rounded.) The measurement of the distance needed to go around the bottle will be the perimeter of the hexagon and the circumference of the circular base of the cylinder. Chapter 5: Solids and Nets 81

50 Solids and Nets cm 6.5 cm The perimeter of the regular hexagon is calculated by multiplying the length of one side by six. P = 3.5 cm 6 = 19.5 cm. The circumference of the circular base is calculated using C = r. The radius of the circle is 3.5 cm. So, C = (3.5 cm) = 6.5 cm. Using 3.14 for, the length of the circumference is: C = 6.5(3.14) cm = 0.41 cm.. The sketch of the nets for the boxes is shown below along with the surface area of each net. The 0.5 cm allowance for space is included in the dimensions. This space allows for the bottle to slide in and out of the box with ease, yet provides minimum movement during shipping. Values are rounded to the nearest hundredth of a centimeter. Cylindrical Box: cm 0.41 cm 8 Chapter 5: Solids and Nets

51 Solids and Nets Hexagonal Box: 5.6 cm 13.5 cm 5.6 cm 19.5 cm 3. The net of the cylindrical box consists of two congruent circles and a rectangle. The area of each circle is A = r. A = (3.5 cm) A = (3.14)(10.56 cm) A = cm The area of the rectangle is A = bh. A = 0.41 cm 13.5 cm A = cm The total surface area of the cylinder is found by adding the areas of the circles and the rectangle: cm cm cm = cm The net of the hexagonal box is comprised of two regular hexagons and a rectangle. The area of the rectangle is A = bh. A = 19.5 cm 13.5 cm A = 63.5 cm The area of the regular hexagon is A = (ap). We know that a =.81 cm and p = 6(3.5 cm) = 19.5 cm A = (.81 cm 19.5 cm) / A = (54.8 cm ) / A = 7.4 cm The total surface area of the hexagonal box is A = 63.5 cm cm cm A = cm Chapter 5: Solids and Nets 83

52 Solids and Nets The hexagonal box has a smaller total surface area of cm, while the cylindrical box has a total surface area of cm. Gina should select the hexagonal box for the perfume because it is more cost-efficient. Extension Question: Suppose the packaging manufacturer used sheets of cardboard 1 meter long and 1 meter wide. How many of each type of packaging could be made from a single sheet? Explain your answer in detail. The area of the cardboard sheet is 1 square meter, or 10,000 square centimeters. You may not, however, divide this total by the total surface area of the box because the shapes, when placed on the sheet, will not use all of the space. The cylindrical shape with tangent circle bases: cm 0.41 cm The total width is 0.41 cm, and the length is , or 6.5 cm. If it is assumed that the shape must be placed with the tangent circles, the question is, How many rectangles of this size will fit in a rectangle 100 cm by 100 cm? An array of three rectangles by four rectangles would fit on the sheet. At least 1 shapes would fit on the sheet. 84 Chapter 5: Solids and Nets

53 Solids and Nets If the circles may be separated from the rectangle, more of the shapes could be cut out of the cardboard. One possible arrangement is shown below. 4 rectangles and 54 circles could be arranged on the square. The total width is 19.5 cm, and the length is or 4.74 cm. The question is, How many rectangles of this size will fit in a rectangle 100 cm by 100 cm? An array of five shapes by four rectangles would fit on the sheet. At least 0 shapes would fit on the sheet. Chapter 5: Solids and Nets 85

54 Solids and Nets If the hexagons are not connected to the rectangles, more could be positioned on the cardboard. One possible arrangement is demonstrated in the diagram below. 8(5.6) = 44.9 cm 3(5.6) = cm 13(5.6) = cm 3(5.6) =16.86 cm Rectangles: 4(5) = 0 (4) = 8 8 rectangles Hexagons: 13(3) = 39 8(3) = 4 63 hexagons Thus, 8 sides and 56 tops and bottoms will fit in the space. 86 Chapter 5: Solids and Nets

55 Solids and Nets Chapter 5: Solids and Nets 87

56 Solids and Nets Student Work A group of four students created a poster of their solution to this problem. The work has been copied on the next three pages. This work exemplifies all of the solution guide criteria. For example: Makes an appropriate and accurate representation of the problem using correctly labeled diagrams. The students diagrams of the solids and nets are correct. They have labeled the measurements of the perfume bottle, the containers, and the nets. Communicates clear, detailed, and organized solution strategy. The students used the words first, and to find these measurements we did the following things. They described what they did and why. They listed the formulas used, what the variables in the formula represented, and what they were for this particular situation. 88 Chapter 5: Solids and Nets

57 Solids and Nets Chapter 5: Solids and Nets 89

58 Solids and Nets 90 Chapter 5: Solids and Nets

59 Solids and Nets Chapter 5: Solids and Nets 91

60 Solids and Nets 9 Chapter 5: Solids and Nets

61 Solids and Nets Playing with Pipes R&B Engineers have been hired to design a piping system for an industrial plant. The design calls for four different-shaped pipes that will carry water to the industrial plant. The pipes will be constructed out of four rectangular pieces of sheet metal, each with a width of 360 cm and a length of 100 cm. The metal will be folded or rolled to make the four pipes, each having cross sections with the same perimeter, 360 cm. One of the pipes has a rectangular cross section with dimensions of 60 cm by 10 cm. One pipe cross section is a square, one is a regular hexagon, and one is circular. The length of each pipe will be 100 cm. 1. Draw a sketch of each pipe, and label the dimensions on each cross section of pipe.. The pipe with the greatest flow is considered to be the pipe with the greatest cross section area. Determine which cross section of pipe will allow for the greatest flow of water, and list them in order from greatest to least. Justify your answers. Chapter 5: Solids and Nets 93

62 Solids and Nets Materials: Notes One graphing calculator per student Geometry TEKS Focus: (G.6) Dimensionality and the geometry of location. The student analyzes the relationship between three-dimensional geometric figures and related two-dimensional representations and uses these representations to solve problems. The student is expected to: (C) use orthographic and isometric views of three-dimensional geometric figures to represent and construct three-dimensional geometric figures and solve problems. (G.8) Congruence and the geometry of size. The student uses tools to determine measurements of geometric figures and extends measurement concepts to find perimeter, area, and volume in problem situations. Teacher Notes Scaffolding Questions: How can the perimeter of the square cross section be determined? The regular hexagonal cross section? The circular cross section? How can you determine the amount of water flow through the pipes? Sample Solutions: 1. All of the pipes have a cross section with the same perimeter. The given perimeter is 360 cm. The dimensions of the rectangular cross section are given in the problem. The dimensions of the other cross sections can be calculated as follows: 360 cm Square: = 90 cm per side cm Regular Hexagon: = 60 cm per side 6 Circle: r = 360 r = 180 r 57.3 cm The pipes are shown below. The student is expected to: (A) find areas of regular polygons, circles, and composite figures; Additional Geometry TEKS: (G.6) Dimensionality and the geometry of location. The student analyzes the relationship between three-dimensional geometric figures and related two-dimensional representations and uses these representations to solve problems Square Rectangle Circular r = 57.3 Regular Hexagon Chapter 5: Solids and Nets

63 Solids and Nets. In order to determine the pipe with the greatest water flow, the area of each cross section must be calculated. Listed in order, from greatest flow to least flow: The area of the circular region will produce an opening that is: A r (57.3) = 10,314 cm. The area of the regular hexagonal region will produce an 1 1 opening that is: A ap cm. The area of the square region will produce an opening that is: A = s = = 8,100 square cm. The area of the rectangular region will produce an opening that is: A = lw = = 7,00 square cm. Extension Questions: The student is expected to: (A) describe and draw the intersection of a given plane with various three-dimensional geometric figures; Connections to TAKS: Objective 7: The student will demonstrate an understanding of two-and three-dimensional representations of geometric relationships and shapes. Objective 8: The student will demonstrate an understanding of the concepts and uses of measurement and similarity. A pipe having a circular cross section with an inside diameter of 10 inches is to carry water from a reservoir to a small city in the desert. Neglecting the friction and turbulence of the water against the inside of the pipes, what is the minimum number of -inch insidediameter pipes of the same length that is needed to carry the same volume of water to the small city in the desert? Explain your answer. In order to solve this problem, it will be necessary to find the number of smaller pipes that it will take to provide the same cross sectional area as that of the larger pipe. The cross sectional areas of the pipes are circles having the area A = r. The radius of the 10-inch pipe is 5 inches, and the radius of the -inch pipe is 1 inch. The area of the 10-inch pipe is A 1 = 5 = 5 in The area of the -inch pipe is A = 1 = 1 in The ratio of the areas is 5 : 1 or 5 : 1. This implies that the cross sectional area of the 10-inch pipe is 5 times larger than the cross sectional area of the smaller pipe. It will require 5 of the -inch pipes to provide the same cross sectional area as the 10-inch pipe. Chapter 5: Solids and Nets 95

64 Solids and Nets An alternate approach is to realize that the ratio of the diameters is 10 to or 5 to 1. Thus, the ratio of the areas is 5 to 1 or 5 to 1. Therefore, the requirement is 5 of the -inch pipes. Suppose the cross sections of the pipes are squares rather than circles. The diagonal of one square is 10 inches, and the diagonal of the second square is inches. Will the shape of the pipes affect the number of pipes required to produce equal areas? Justify your answer. It makes no difference what shapes the pipes are, provided that the cross sectional areas are the same. If the pipes had square cross sections, and the diagonals were 10 inches and inches, it will still require 5 smaller pipes to provide the same cross sectional area as the larger pipe. The pipe with the 10-inch diagonal forms two triangles with leg lengths of 5 inches. Therefore, the square cross section will have side lengths of 5 inches. The area of the pipe with the 10-inch diagonal will be A 5 50 in. The pipe with the -inch diagonal forms two triangles with leg lengths of 1 inches. Therefore the square cross section will have side lengths of 1 inches. The area of the pipe with the -inch diagonal will be A 1 50 in. The ratio of the areas of the square cross section is 50 : or 5: 1. Therefore, it will require 5 of the smaller square pipes to provide the same cross sectional area as the larger pipe. 96 Chapter 5: Solids and Nets

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