Math 6: Geometry 3-Dimensional Figures Three-Dimensional Figures A solid is a three-dimensional figure that occupies a part of space. The polygons that form the sides of a solid are called a faces. Where the faces meet in segments are called edges. Edges meet at vertices. A prism is a solid formed by polygons. The faces are rectangles. The bases are congruent polygons that lie in parallel planes. A pyramid is a solid whose base may be any polygon, with the other faces triangles. vertex base Polyhedra are solids with all faces as polygons. Prisms and pyramids would meet this criterion, while cylinders and cones would not, therefore they will be discussed at a later time. A picture is worth a thousand words. The ability to draw three-dimensional figures is an important visual thinking tool. some drawing tips: Here are Rectangular Prism (face closest to you): Draw the front rectangle. Draw a congruent rectangle in another position. Connect the corners of the rectangles. Use dashed lines to show the edges you would not see. Your rectangular prism! Rectangular Prism (edge closest to you): Math 6 Notes Geometry: 3-Dimensional Figures Page 1 of 43
6.G.A.2 Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = B h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems. Volume If you were to buy dirt for your yard, it s typically sold in cubic yards that s describing volume. If you were laying a foundation for a house or putting in a driveway, you d want to buy cement, and cement is often sold by the cubic yard. Carpenters, painters and plumbers all use volume relationships. 1 m 1 m 3 m 2 m The volume of a three dimensional figure measures how many cubes will fit inside it. It s easy to find the volume of a solid if it is a rectangular prism with whole number dimensions. Let s consider a figure 3 m 2 m 4 m. We can count the cubes measuring 1 meter on an edge. The bottom layer is 3 2 there are 6 square meter cubes on the bottom layer. We have three more layers stacked above it (for a total of 4 layers), or 6 + 6 + 6 + 6 = 24. 4 m 3 m 2 m Now we can reason that if I know how many cubes are in the first layer (6), then to find the total number of cubes in the stack, you simply multiply the 6 4 = 24. number on the first layer by the height of the stack ( ) This is a way of finding volume. We find the area of the base (B) and multiply it times the height (h) of the object. For prisms, V = Bh, where B is the area of the base and h is the height. Since rectangular prisms have bases that are rectangles, B = A rectangle = lw. Therefore, we use the formula V = lwh. The answer in a volume problem is always given in cubic units (cm 3, in 3, ft 3, ) because we are determining how many cubes will fill the solid. Example: Julie is using sugar cubes to create a model for a school project; each sugar cube has an edge length of 1 cm. After building her first model, she realizes that she must increase each measure by 1½ times. The diagram given shows her first try. Math 6 Notes Geometry: 3-Dimensional Figures Page 2 of 43
4 m 2 m 3 m a) How many cubes did she use to build her first model? V = lwh V = V = 24 ( 3)( 2)( 4) The volume is 24 cm 3. b) What would the new dimensions be after she increases each measure? Original Measure Rate of Change New Measure (slope) & (dilations) Length = 3 cm 1½ 3 1½ = 4½ cm Width = 2 cm 1½ 2 1½ = 3 cm Height = 4 cm 1½ 4 1½ = 6 cm c) How many more sugar cubes will she need to complete her project? 4 m 3 m 2 m First, she must find the volume of the larger model, then subtract the volume of the original model to find the number of cubes she will need to finish. This could be done physically by adding two blocks on top (height), 1 block behind (width) and 1½ blocks to the side (length). OR Just use the formula V = lwh 1 V = 4 2 V = 81 ( 3)( 6) The larger volume is 81 cm 3. 81-47=57, so she needs 57 more sugar cubes. Example: The diagram below shows a cube with sides of length 30cm. A smaller cube with side length 5 cm has been cut out of the larger cube. a) What is the volume of the large cube before the small cube is cut out? TOTAL SURFACE AREA = sum of all faces The volume of the SA = 400 + 780 + 260 + 360 + 342 + 342 large cube is 27,000 cm 3. V = 2484 Math 6 Notes Geometry: 3-Dimensional Figures Page 3 of 43
b) What is the volume of the small cube being cut out? V = lwh V = ( 5)( 5)( 5) V = 125 The volume of the small cube is 125 cm 3. c) What is the volume of the solid left? V = lwh TOTAL VOLUME = V large cube - V TOTAL VOLUME = 27, 000 125 TOTAL VOLUME = 26,875 small cube The total volume of the solid is 26,875 cm 3. Example: If the volume of the rectangular prism is 450,000 cm 3, the value of x is a) 0.06 m b) 0.6 m c) 6 m d) 60 m Note the units in the question and the units used in the answer choices. We need to convert so we are comparing like measurements. Every meter contains 100 centimeters 3 so a cubic centimeter would measure 100cm 100cm 100cm or 1,000,000cm, therefore the volume into m 3 (converting using ratios) V = lwh 3 1m 3 3 450, 000cm = 0.45m and 0.45 = 3 ( 1.5)( 0.5)( x). 1,000,000cm 0.45 = 0.75x Students can guess and check to find the answer. Some number sense can make the answer almost obvious. 0.45 is a little more than half of 0.75, so to find the answer 0.75 must be multiplied by a little more than half. Only one choice fits that criterion (b) Example: By how much will the volume of a rectangular prism increase, if its length, width, and height are doubled? a) 4 times b) 2 times c) 6 times d) 8 times Students can use a strategy of looking for a pattern Math 6 Notes Geometry: 3-Dimensional Figures Page 4 of 43
V = lwh Volume of a 1 1 1 cube V = ( 1)( 1)( 1) = V = 1 now double the measures V = lwh Volume of a 2 2 2 cube V = ( 2)( 2)( 2) = V = 8 3 1unit, 3 8units WOW, 8 times as much. (d) Example: By how much will the volume of a rectangular prism increase, if its length is doubled? a) 2 times b) 4 times c) 8 times d) 6 times Again, students can look for a pattern V = lwh Volume of a 1 1 1 cube V = ( 1)( 1)( 1) = V = 1 now double the length measure V = lwh Volume of a 2 2 2 cube V = ( 2)( 1)( 1) = V = 2 3 1unit, 3 2units 2 times as much this time. (a) Surface Area 6.G.A.4 Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems. Another way to look at three-dimensional figures is to look at a net. A net is an arrangement of two-dimensional figures that can be folded to make three-dimensional figures. This will take the student from two-dimensions to three-dimensions. Also have students start working the other way: start with a three-dimensional solid, like a box, and see if they can draw what it would look like if it was unfolded and laid flat. Students could even cut out their drawing and try to recreate the solid. The following websites will give you more resources: http://www.mathisfun.com/platonic_solids.html printable nets for the platonic solids, shows figures rotating (cube and tetrahedron only) http://www.senteacher.org/wk/3dshape.php printable nets for many different solids Math 6 Notes Geometry: 3-Dimensional Figures Page 5 of 43
In addition to drawing solid figures and working with nets, students are expected to create twodimensional drawings of three-dimensional figures and create three dimensional figures from a two-dimensional drawing. For these notes and the creating of the practice test and test, we have used Microsoft Word. Choose Insert Shapes then choose the cube in the Basic Shapes section. You are then able to stack and build almost any 3-D shape of your choosing. Once your figure is built you can group the figure to lock the shape. In class you can have students build 3- D figures using wooden cubes, stacking cubes, interlocking cubes or Lego pieces to develop the ability to see the top view, side view and front view. Example: Given the following figure, identify (or draw) the top view, side view and front view. From the top view, you would see From the front view, you would see From the side view, you would see Example: Given the following figure, identify (or draw) the top view, side view and front view. From the top view, you would see From the front view, you would see From the side view, you would see Allow students to build and draw figures. As always, begin with very simple figures and allow them to try more complex figures as they are able. Math 6 Notes Geometry: 3-Dimensional Figures Page 6 of 43
Example: Given the following figure, identify (or draw) the top view, side view and front view. From the top view, you would see From the front view, you would see From the side view, you would see Example: Given the top, side and front views, identify (or draw) the figure. Top View Front View Side View Answer: Example: Given the top, side and front views, identify (or draw) the figure. Top View Front View Side View Answer: Math 6 Notes Geometry: 3-Dimensional Figures Page 7 of 43
Example: Given the top, side and front views, identify (or draw) the figure. Top View Front View Side View Answer: Example: Given the top, side and front views, identify (or draw) the figure. Top View Front View Side View Answer: Surface Area The surface area of a solid is the sum of the areas of all the surfaces that enclose that solid. To find the surface area, draw a diagram of each surface as if the solid was cut apart and laid flat. Label each part with the dimensions. Calculate the area for each surface. Find the total surface area by adding the areas of all of the surfaces. If some of the surfaces are the same, you can save time by calculating the area of one surface and multiplying by the number of identical surfaces. Remind your students that nets are a way to break up these figures into surfaces for which we can easily find the area. Try to have students imagine the process of unfolding Math 6 Notes Geometry: 3-Dimensional Figures Page 8 of 43
The following shows an example of the net of a triangular prism Example: Which of the following is NOT the net of a pyramid? a) c) b) d) Answer: (b) Example: Find the surface area of the prism shown. All surfaces are squares. 7 cm Divide the prism into its parts. Label the dimensions. Bases Lateral Faces top bottom back front side side 7 cm 7 cm 7 cm 7 cm 7 cm 7 cm Math 6 Notes Geometry: 3-Dimensional Figures Page 9 of 43
Find the area of all the surfaces. Bases Lateral Faces A = bh A= 7 7 A = 49 A = bh A= 7 7 A = 49 A = bh A= 7 7 A = 49 A = bh A= 7 7 A = 49 A = bh A= 7 7 A = 49 A = bh A= 7 7 A = 49 Surface Area = Area of the top + bottom + front + back + side + side Surface Area = 49 + 49 + 49 + 49 + 49 + 49 = 294 The surface area of the prism is 294 cm 2. Since a cube has 6 congruent faces, a simpler method would look like Surface Area = 6 the area of a face Surface Area = 6B Surface Area = 6bh = 677 = 42 7 = 294 Again, the surface area of the prism is 294 cm 2. Example: Find the surface area of the prism shown. All surfaces are rectangles. 4 cm 15 cm Divide the prism into its parts. Label the dimensions. 2 cm Bases Lateral Faces 2 top 15 4 front 15 4 side 2 2 bottom 15 4 back 4 side 15 2 Math 6 Notes Geometry: 3-Dimensional Figures Page 10 of 43
Find the area of all the surfaces. Bases A = bh A = 15 2 A = 30 A = bh A = 15 2 A = 30 A = bh A = 15 4 A = 60 A = bh A = 15 4 A = 60 Lateral Faces A = bh A = 2 4 A = 8 A = bh A = 2 4 A = 8 Surface Area = Area of the top + bottom + front + back + side + side Surface Area = 30 + 30 + 60 + 60 + 8 + 8 = 196 The surface area of the prism is 196 cm 2. Note: Since some of the faces were identical, we could multiply by 2 instead of adding the value twice. That work would look like Surface Area = 2(top or bottom) +2(front or back) +2(side) Surface Area = 2(30) + 2(60) + 2(8) = 60 + 120 + 16 = 196 Again, the surface area of the prism is 196 cm 2. Example: Find the surface area of the triangular prism. If we break our triangular prism down into a net, we get this: In a triangular prism there are five faces, two triangles and three rectangles. The total surface area would be the sum of all the areas 1 A 2 b triangle = h 1 A = 2 A = 18 ( 9.0)( 4) Math 6 Notes Geometry: 3-Dimensional Figures Page 11 of 43
TOTAL SURFACE AREA = sum of the areas of all faces SA = 36.45 + 72.9 + 58.32 + 18 + 18 SA = 203.67 The surface area is 203.67 cm 2. Example: Find the surface area of the trapezoidal prism. A net of this solid would look something like this Filling in measurements With the both bases 20 X 20 = 400 39 X 20 = 780 13 X 20 = 260 18 X 20 = 360 The total surface area would be the sum of all the areas TOTAL SURFACE AREA = sum of the areas of all faces SA = 400 + 780 + 260 + 360 + 342 + 342 SA = 2484 The surface area is 2484 m 2. 1 A= ( b 1+b2 ) h 2 1 A = 18 + 39 12 2 1 A = ( 57 )( 1 2) 2 A = 342 ( )( ) Math 6 Notes Geometry: 3-Dimensional Figures Page 12 of 43
Example: Find the surface area of the isosceles trapezoidal prism. 5 5 5 5 5 4 11 5 3 The total surface area would be the sum of all the areas TOTAL SURFACE AREA = sum of the areas of all faces SA = 15 + 15 + 15 + 34 + 32 + 32 SA = 143 The surface area is 143 in 2. 1 A= ( b 1+b2 ) h 2 1 A = 5 + 11 4 2 1 A = ( 16 )( 4) 2 A = 32 ( )( ) Example: Find the surface area of the triangular prism. 3 8 8 8 12 The total surface area would be the sum of all the areas TOTAL SURFACE AREA = sum of the areas of all faces SA = 96 + 96 + 96 + 12 + 12 SA = 312 The surface area is 312 cm 2. 1 A triangle = bh 2 1 A = 2 A = 12 ( 8)( 3) Math 6 Notes Geometry: 3-Dimensional Figures Page 13 of 43
Example: Find the surface area of the rectangular pyramid. 12 10 1 = 2 b h 1 A = 12 10 2 A = 60 A triangle ( )( ) 12 8 a) 206 m 2 b) 312 m 2 c) 302 m 2 d) 216 m 2 1 A triangle = bh 2 1 A = 2 A = 48 ( 8)( 12) The total surface area would be the sum of all the areas TOTAL SURFACE AREA = sum of the areas of all faces SA = 96 + 48 + 48 + 60 + 60 SA = 312 The surface area is 312 m 2. Example: The surface area of the composite solid of the figure below is a) 5000 cm 2 b) 4950 cm 2 c) 4550 cm 2 d) 4450 cm 2 Math 6 Notes Geometry: 3-Dimensional Figures Page 14 of 43
The net for this solid consists of six rectangles and two oddly shaped bases. 5 20 5 5 25 15 30 5 20 5 X 5 = 25 30 X 15 = 450 Decomposed Area of base = 450 + 25 = 475 40 40 X 20 = 800 40 X 25 = 1000 40 X 5 = 200 40 X 5 = 200 40 X 15 = 600 40 X 30 = 1200 The total surface area would be the sum of all the areas TOTAL SURFACE AREA = sum of the areas of all faces SA = 800 + 200 + 200 + 1000 + 600 + 1200 + 60 + 475 + 475 SA = 5010 The surface area is 5010 cm 2. Optional Extension: The formula for Total Surface Area of a rectangular prism is given as: SA = 2lw +2wh+2lh In addition, a discussion about the difference between Total Surface Area and Lateral Area can be introduced. Lateral Area is defined to be the surface area of a threedimensional object minus the area(s) of the base(s). We can call the faces included in the Lateral Area the lateral faces. Can you find the Lateral Area of the solid in the previous example? Math 6 Notes Geometry: 3-Dimensional Figures Page 15 of 43
Notice the absence of the two bases in the following diagram. 20 5 5 25 15 30 40 The sum of the areas of the six rectangles would be the Lateral Surface Area. It can be found by adding the areas of each lateral face LATERAL AREA = sum of the areas of the lateral faces LA = 800 + 200 + 200 + 1000 + 600 + 1200 + 60 LA = 4060 OR by taking the Total Surface Area and subtracting the areas of the two bases LATERAL AREA = TOTAL SURFACE AREA minus of the areas of the bases LA = 5010 (475 + 475) LA = 4060 The Lateral Area is 4060 cm 2. Example: An area to be planted with grass seed measures 50 feet by 75 feet. Before planting, a 3-inch layer of loam is spread on the area. Part A: How many cubic feet of loam is needed? Part B: A truck delivers loam in cubic yards. The landscaper divides the cubic feet of loam is needed by 9 to find the cubic yards that will be needed. Will this calculation produce the correct results? Explain your answer. Part C: How many cubic yards of loam will need to be delivered? Answer: Part A: Since the units are not the same they must be converted. We can review the ratio conversions in this process. The loam would create a rectangular prism on top of the garden that is 50 ft 75 ft 3in, either both 50 ft and 75 ft must be converted or 3 inches must be converted. Let s see what happens both ways 50 ft 12 in = 600 in 1 1ft AND 75 ft 12 in = 900 in 1 1ft So the new dimensions are 600in 900in 3in Math 6 Notes Geometry: 3-Dimensional Figures Page 16 of 43
Using those dimensions we can find the volume in cubic inches. V = lwh The volume is 1,620,000 in 3 which must be converted back into cubic feet. V = ( 600)( 900)( 3) V = 1,620,000 MISCONCEPTION: Since each foot contains 12 inches students want to divide by 12 to obtain the answer. WRONG!! 1 cubic foot V = lwh V = V = 1 ( 1)( 1)( 1) 1,728 cubic inches V = lwh V = V = 1,728 ( 12)( 12)( 12) They must actually divide by 1,728!! That s a BIG difference! 1 ft. 12 in. 1,620,000 1728 = 937. 5 937.5 ft 3 of loam will be needed. OR 3in 1 ft 3 1 = ft = ft 1 12 in 12 4 1 So the new dimensions are 50 ft 75 ft in. 4 V = lwh 1 V = ( 50)( 75) 4 V = 937.5 The volume is 937.5 ft 3. Notice, in this case, no further conversions are necessary. Nice convenience, but we did have to work with fractions. Part B: Again, watch for that misconception when converting. The correct conversion is shown below. 1 cubic yard 27 cubic feet V = lwh V = V = 1 ( 1)( 1)( 1) V = lwh V = V = 27 ( 3)( 3)( 3) 1 yd 3 ft So NO, the landscaper is wrong! He will order too much, because a cubic yard is 27 cubic feet he needs to divide by 27. Part C: 1 27 ft 27 3 3 937.5 ft 1 yd 937.5 3 3 3 = yd = 34.72 yd 35 yd 3 35 cubic yards of loam will be delivered. Math 6 Notes Geometry: 3-Dimensional Figures Page 17 of 43
Sample Explorations Questions Correct Answer: A Correct Answer: D Correct Answer: C Math 6 Notes Geometry: 3-Dimensional Figures Page 18 of 43
Correct Answer: B Correct Answer: B Correct Answer: D Correct Answer: B Math 6 Notes Geometry: 3-Dimensional Figures Page 19 of 43
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Correct Answer: C Correct Answer: C Math 6 Notes Geometry: 3-Dimensional Figures Page 22 of 43
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Correct Answer: B Correct Answer: C Correct Answer: A Correct Answer: A Math 6 Notes Geometry: 3-Dimensional Figures Page 25 of 43
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Correct Answer: V= 54 ft 2 SA=102 ft 2 Math 6 Notes Geometry: 3-Dimensional Figures Page 32 of 43
2013 Sample SBAC Questions Standard: 6.G.1, 6.G.3 DOK: 2 Item Type: TE Difficulty: M Math 6 Notes Geometry: 3-Dimensional Figures Page 33 of 43
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Standard: 6.G.2, 6.NS.3 DOK: 2 Item Type: ER Difficulty: M Math 6 Notes Geometry: 3-Dimensional Figures Page 35 of 43
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Standard: 6.G.4 DOK: 2 Item Type: TE Difficulty: M Math 6 Notes Geometry: 3-Dimensional Figures Page 37 of 43
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2014 Sample SBAC Questions Math 6 Notes Geometry: 3-Dimensional Figures Page 39 of 43
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This example is a repeat from 2013. Math 6 Notes Geometry: 3-Dimensional Figures Page 42 of 43
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