Math 6: Geometry Notes 2-Dimensional and 3-Dimensional Figures

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1 Math 6: Geometry Notes 2-Dimensional and 3-Dimensional Figures Short Review: Classifying Polygons A polygon is defined as a closed geometric figure formed by connecting line segments endpoint to endpoint. Polygons Not Polygons Polygons are named by the number of sides. We know a triangle has 3 sides. Below are the names of other polygons. Polygons Quadrilateral Pentagon Hexagon Heptagon Octagon Nonagon Decagon # of sides Short Review: Classifying Triangles Triangles can be classified by the measures of their angles: acute triangle 3 acute angles right triangle 1 right angle obtuse triangle 1 obtuse angle Example: Classify each triangle by their angle measure: M A 20 D G E B C F H J K Acute (Equiangular) Right Obtuse Acute L Triangles can also be classified by the lengths of their sides. You can show tick marks to show congruent sides. equilateral triangle 3 congruent sides isosceles triangle at least 2 congruent sides scalene triangle no congruent sides Math 6 Notes Geometry: 2-Dimensional and 3-Dimensional Figures Page 1 of 46

2 equilateral isosceles scalene Example: Classify the triangle. The perimeter of the triangle is 15 cm. 10 cm Using the information given regarding the perimeter: 2.5 cm x x x x 2.5 Since 2 sides are congruent, the triangle is isosceles. A tree diagram could also be used to show the triangle relationships. Tree Diagram for Triangles triangles acute obtuse right scalene isosceles scalene isosceles scalene isosceles equilateral Short Review: Classifying Quadrilaterals A quadrilateral is a plane figure with four sides and four angles. They are classified based on congruent sides, parallel sides and right angles. Quadrilateral Type Definition Example >> Parallelogram Quadrilateral with both pairs of opposite sides parallel. >> Rhombus Parallelogram with four congruent sides. Note: This polygon is a parallelogram. Rectangle Parallelogram with four right angles. Note: This polygon is a parallelogram. Math 6 Notes Geometry: 2-Dimensional and 3-Dimensional Figures Page 2 of 46

3 Square Parallelogram with four right angles and four congruent sides. Note: This polygon is a parallelogram. Trapezoid Quadrilateral with exactly one pair of parallel sides. >> >> Another way to show the relationship of the parallelograms is to complete a Venn diagram as shown below. Parallelograms Rectangles Squares Rhombi Vocabulary becomes very important when trying to solve word problems about quadrilaterals. Example: A quadrilateral has both pairs of opposite sides parallel. One set of opposite angles are congruent and acute. The other set of angles is congruent and obtuse. All four sides are NOT congruent. Which name below best classifies this figure? A. parallelogram B. rectangle C. rhombus D. trapezoid We have both pairs of opposite sides parallel, so it cannot be the trapezoid. Since the angles are not 90 in measure, we can rule out the rectangle. We are told that the 4 sides are not congruent, so it cannot be the rhombus. Therefore, we have a parallelogram. (A) 6.G.A.1 Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems. Math 6 Notes Geometry: 2-Dimensional and 3-Dimensional Figures Page 3 of 46

4 Area of Triangles and Quadrilaterals One way to describe the size of a room is by naming its dimensions. A room that measures 12 ft. by 10 ft. would be described by saying it s a 12 by 10 foot room. That s easy enough. There is nothing wrong with that description. In geometry, rather than talking about a room, we might talk about the size of a rectangular region. For instance, let s say I have a closet with dimensions 2 feet by 6 feet (sometimes given as 2 6). That s the size of the closet. 2 ft. Someone else might choose to describe the closet by determining how many one foot by one foot tiles it would take to cover the floor. To demonstrate, let me divide that closet into one foot squares. 6 ft. 2 ft. By simply counting the number of squares that fit inside that region, we find there are 12 squares. If I continue making rectangles of different dimensions, I would be able to describe their size by those dimensions, or I could mark off units and determine how many equally sized squares can be made. Rather than describing the rectangle by its dimensions or counting the number of squares to determine its size, we could multiply its dimensions together. Putting this into perspective, we see the number of squares that fits inside a rectangular region is referred to as the area. A shortcut to determine that number of squares is to multiply the base by the height. 6 ft. The area of a rectangle is equal to the product of the length of the base and the length of a height to that base. That is. Most books refer to the longer side of a rectangle as the length (l), the shorter side as the width (w). That results in the formula. The answer in an area problem is always given in square units because we are determining how many squares fit inside the region. Math 6 Notes Geometry: 2-Dimensional and 3-Dimensional Figures Page 4 of 46

5 height height Example: Find the area of a rectangle with the dimensions 3 m by 2 m. A 32 The area of the rectangle is 6 m 2. Example: Find the area of the rectangle. 9 ft. 2 yd. Be careful! Area of a rectangle is easy to find, and students may quickly multiply to get an answer of 18. This is wrong because the measurements are in different units. We must first convert feet into yards, or yards into feet. yards 1 x feet 3 9 We now have a rectangle with dimensions 3 yd. by 2 yd. 9 3x 3 x A 32 The area of our rectangle is 6 square yards. If I were to cut one corner of a rectangle and place it on the other side, I would have the following: base base A parallelogram! Notice, to form a parallelogram, we cut a piece of a rectangle from one side and placed it on the other side. Do you think we changed the area? The answer is no. All we did was rearrange it; the area of the new figure, the parallelogram, is the same as the original rectangle. This allows us to find a formula for the area of a parallelogram. Math 6 Notes Geometry: 2-Dimensional and 3-Dimensional Figures Page 5 of 46

6 Since the bottom length of the rectangle was not changed by cutting, it will be used as the base length (b), the height of the rectangle was not changed either, we ll call that h. h Now we arrive at the formula for the area of a parallelogram.. b Example: The height of a parallelogram is twice the base. If the base of the parallelogram is 3 meters, what is its area? First, find the height. Since the base is 3 meters, the height would be twice that or 2(3) or 6 m. To find the area, A 36 A 18 The area of the parallelogram is 18 m 2. We have established that the area of a parallelogram is understand the area formula for a triangle and trapezoid.. Let s see how that helps us to Remember: Once a formula for a figure has been developed, it can be used for any figure that meets its criteria. For example: The parallelogram formula can be used for rectangles, rhombi, and squares. The rectangle formula can be used for squares. The rhombus formula (derived in HS Geometry) can be used for squares. This is based on the Venn Diagram given previously (pg. 3 of these notes). The inner sets have all the same attributes and properties of the sets they are contained within. Therefore, what must be true about any element of the outer set must be true of all elements of that set. Math 6 Notes Geometry: 2-Dimensional and 3-Dimensional Figures Page 6 of 46

7 h base h base For this parallelogram, its base is 4 units and its height is 3 units. Therefore, the area is. If we draw a diagonal, it cuts the parallelogram into 2 triangles. That means one triangle would have one-half of the area or 6 units 2. Note the base and height stay the same. So for a triangle, h base h base For this parallelogram, its base is 8 units and its height is 2 units. Therefore, the area is. If we draw a line strategically, we can cut the parallelogram into 2 congruent trapezoids. One trapezoid would have an area of one-half of the parallelogram s area (8 units 2 ). Height remains the same. The base would be written as the sum of. For a trapezoid: Composite Figures are figures made up of multiple shapes. (linkage - Composite numbers have multiple factors) In order to find the area of these oddly-shaped figures they must be decomposed into figures we are familiar with. Let s start with the trapezoid 10 in. 4 in. 6 in. Of course the trapezoid formula can be used but we can also decompose this trapezoid into a rectangle and two triangles. The area of this trapezoid would be the area of the rectangle added to the areas of the two triangles. Math 6 Notes Geometry: 2-Dimensional and 3-Dimensional Figures Page 7 of 46

8 In this case, because the trapezoid is isosceles, the two triangles will be congruent Notice the b 1 length 10 is equal to the b 2 length 6, plus 4 more inches. Those 4 inches can be split into 2 and 2 and would indicate the lengths of the bases of the two triangles. 4 in. 10 in. 6 in. Now we can use the rectangle formula and triangle formula (twice) to find the total area. Rectangle A A Triangle A 2 2 A in. 4 in. 6 in. 6 in. 2 in. Since the two triangles are congruent, 24 A 24 The area of this trapezoid is 32 square A 32 inches. By applying the trapezoid formula we can check to see if our answer is correct. 1 A A A 32 Once again, we see the area of the trapezoid is 32 in 2. Now that we see that it can be done, we can explore the areas of other, less common, composite shapes. These shapes must be decomposed and the area formulas of the decomposed parts can be used to find each individual area. The total area of the figure is the sum of the areas of its decomposed parts. Math 6 Notes Geometry: 2-Dimensional and 3-Dimensional Figures Page 8 of 46

9 Example: Find the area of the given shape. A B C Start by decomposing 7cm-5cm = 2cm One way to decompose is given above. Students may find multiple ways to decompose the given figures. If the needed measurements are not available for them to complete the problem, they may want to consider trying a different combination. (Re-decompose?) From the diagram we can see that the total area of this figure will be the sum of the areas of the three rectangles that composed it. Notice that the measures of the sides of the figure can be found by counting the blocks that run along the side of that portion. (Be careful of the scale of the diagram, sometimes a block represents more than one unit.) TOTAL A A A 45 The area of this figure is 45 square cm. Students can verify this answer by counting the squares inside the figure. Example: Find the area of the given polygon. (DOK 2) First, the figure must be decomposed Students should be able to find a rectangle and a triangle. TOTAL A. 102 m 2 B. 187 m 2 C. 289 m 2 D. 391 m 2 remember... lw A A A 289 and 1 bh 2 So, the area of this figure is 289 m 2. (C) Math 6 Notes Geometry: 2-Dimensional and 3-Dimensional Figures Page 9 of 46

10 Example: Find the area of the given polygon. (DOK 2) A. 45 cm 2 B. 90 cm 2 C. 110 cm 2 D. 180 cm 2 This figure can be decomposed a few ways: Vertically down the middle, forming two congruent triangles: b = = 15 cm, h = 12 2 = 6 cm Therefore, the total area is twice the area of one triangle. 1 bh bh 2 The total area is 90 cm 2. (b) 1 ATOTAL A 90 Another way would be: Horizontally, along the diagonal, forming two non-congruent triangles: b = 12 cm (on both), h 1 = 4 cm (top ) & h 2 = 11 cm (bottom ) In this case we must find the sum of both areas to find the total sum. A A TOTAL TOTAL A 90 1 bh The total area is 90 cm 2. (b) Math 6 Notes Geometry: 2-Dimensional and 3-Dimensional Figures Page 10 of 46

11 A final alternative would be: Horizontally and vertically, forming four right triangles, with two sets congruent: (these measurements are interchangeable based on your perspective when viewing the triangles) b 1 = 4 cm (both smaller s) & b 2 = 11 cm (both larger s), h = 12 2 = 6 cm This time we must add the areas of all four triangles together, but recall that there were two sets of congruent triangles formed when we decomposed the original figure in this manner. So We can find the area of one small triangle and double it, then find the area of one larger triangle and double it, and finally add those two doubled areas together. 1 bh A A TOTAL TOTAL A Example: Find the area of the given polygon. Taa daa The total area is 90 cm 2. (b) AGAIN!! The area of this figure can be found multiple ways. It could be decomposed into rectangles this way TOTAL A A A The total area is cm 2. An alternative to this method and extension on this topic is to decompose and subtract. In this case, students can picture an imaginary rectangle surrounding the entire figure then subtract the region that is not a part of the original figure. Math 6 Notes Geometry: 2-Dimensional and 3-Dimensional Figures Page 11 of 46

12 TOTAL A A A Again, the total area is cm 2. Extensions Examples: Find the areas of the shaded regions in the figures below. a) All squares are rectangles... TOTAL A A 16 4 A 12 The area of the shaded region is 12 square cm. b) To find the area of the inner region, we must decompose it. A inner region A 8 4 A Now the shaded region can be found. A A 60 The area of the shaded region is 60 square cm. 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 Math 6 Notes Geometry: 2-Dimensional and 3-Dimensional Figures Page 12 of 46

13 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. Here are some drawing tips: 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): 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. Math 6 Notes Geometry: 2-Dimensional and 3-Dimensional Figures Page 13 of 46

14 1 m 3 m 1 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 = 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 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,, where B is the area of the base and h is the height. Since rectangular prisms have bases that are rectangles,. Therefore, we use the formula. 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. For testing at the state level (grade 6) on volume, solid figures may only include cubes or right rectangular prisms. For that reason, the focus for volume should be. 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. a) How many cubes did she use to build her first model? 4 m 2 m 3 m V V 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 31½ = 4½ cm Width = 2 cm 1½ 21½ = 3 cm Height = 4 cm 1½ 41½ = 6 cm Math 6 Notes Geometry: 2-Dimensional and 3-Dimensional Figures Page 14 of 46

15 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 V The larger volume is 81 cm =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? SA V 2484 The volume of the large cube is 27,000 cm 3. b) What is the volume of the small cube being cut out? V V The volume of the small cube is 125 cm 3. c) What is the volume of the solid left? 27, ,875 The total volume of the solid is 26,875 cm 3. Math 6 Notes Geometry: 2-Dimensional and 3-Dimensional Figures Page 15 of 46

16 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) 3 1m 1,000,000 cm , 000cm 0.45m and x x Students can guess and check to find the answer. Some number sense can make the answer almost obvious 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 Volume of a cube V V 1 now double the measures 111 = 3 1unit,. Volume of a 22 2 cube V V = 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 Math 6 Notes Geometry: 2-Dimensional and 3-Dimensional Figures Page 16 of 46

17 Again, students can look for a pattern Volume of a cube V V 1 now double the length measure 111 = 3 1unit, Volume of a 22 2 cube V V = 3 2units 2 times as much this time. (a) 6.G.A.3 Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems. Example: Find the length of segment CD. (DOK 1) Simply count the units along the indicated side. Common error Remind students to start at 0 and count each swoop as 1 unit CD = 3 units Example: In each part below two sides of a rectangle are shown. Write the coordinates of the fourth corner of each rectangle. Then answer the questions. a) The fourth corner is at (5, -5). Can the perimeter and area of this rectangle be found? If so, what are they? If not, why not? Yes, they are P A 10 P A 90 P 38 The perimeter is 38 units. The area is 90 units 2. and 9 Math 6 Notes Geometry: 2-Dimensional and 3-Dimensional Figures Page 17 of 46

18 b) The fourth corner is at (5, 0). Can the perimeter and area of this rectangle be found? If so, what are they? If not, why not? NO, because the sides are not horizontal or vertical so the measures cannot be determined. Example: Which gives the perimeter of the square? (DOK 2) By counting the units we can see that l = 3 & w = 3, therefore P P 66 P 12 OR Since the side lengths of a square are always equal, it may be faster for students to find one side length and multiply by 4. a) 4 c) 10 b) 8 d) 12 P 12 P 43 Both methods yield the answer 12 units. (d) Example: Plot the sixteen points in the table below on this graph. After graphing the points, connect them to make a 16-pointed star. POINTS POINTS POINTS POINTS A(4, 0) E(-4, 0) I(0, 4) M(0, -4) B(1, 2) F(1, -2) J(3, 3) N(3, -3) C(2, 1) G(2, -1) K(-1, 2) P(-1, -2) D(-3, 3) H(-3, -3) L(-2, 1) Q(-2, -1) Math 6 Notes Geometry: 2-Dimensional and 3-Dimensional Figures Page 18 of 46

19 E D L K I B C J A Q P H F M G N a) Find all horizontal lengths: DJ = 6 units KB = 2 units LC = 4 units EA = 8 units QG = 4 units PF = 2 units HN = 6 units b) Find all vertical lengths: DH = 6 units LQ = 2 units KP = 4 units IM = 8 units BF = 4 units CG = 2 units JN = 6 units c) Find the area of the triangle found by connecting the vowels ( AEI) 1 bh 2 1 A 84 2 The area of AEI is 16 square units or 16 units 2. A 16 d) Discussion: I purposely skipped labeling any point with an O, what point is usually labeled with an O? Give its name and coordinates. The origin, (0, 0) Example: In each question below the coordinates of three corners of a square are given. Find the coordinates of the other corner in each case. You may find it helpful to draw a sketch. a) (2, -2), (2, 3) and (-3, 3). The other corner of the square is at (, ). (-3, -2) b) (2, 3), (3, 4) and (1, 4). The other corner of this square is at (, ). (2, 5) c) (2, 2), (4, 4) and (4, 0). The other corner of this square is at (, ). (6, 2) Math 6 Notes Geometry: 2-Dimensional and 3-Dimensional Figures Page 19 of 46

20 Answers: a) b) c) 5 units 5 units Example: The line marked on the coordinate grid below is one side of a square: What are the possible coordinates of the corners of the square? There are two possible places the square could be placed. Students can visualize (or sketch) the imaginary right triangle with the given line as the hypotenuse. By using it as a reference, students can then find the location of the missing vertices of the square. LINKAGE: slope, rate of change The missing coordinates could be at (6, 3) and (1, 6) OR (-5, -4) and (0, -7). Math 6 Notes Geometry: 2-Dimensional and 3-Dimensional Figures Page 20 of 46

21 Example: Two corners of a square are shown in the coordinate plane below: a) If the third corner is at (7, -1), where is the fourth corner? (5, 3) b) If the third corner is at (-3, -1) where is the fourth corner? (-1, -5) It is possible to make a different square to those above by placing the third and fourth points in two new positions. c) What are the coordinates which need to be plotted? By looking at the given points as opposite vertices, we can find the other corners at (4, 0) and (0, -2). Example: Identify the coordinates of vertex D after quadrilateral DEFG is translated 7 units up: (DOK 2) A translation up will change the y-coordinate only and move the figure up seven units. Therefore, D(3, -2) would move to (3, 5). (b) a) (3, -2) b) (3, 5) c) (5, 3) d) (-4, -2) Math 6 Notes Geometry: 2-Dimensional and 3-Dimensional Figures Page 21 of 46

22 Example: John rode his bike on Monday and again on Friday. His path for Monday is shown on the graph below. Each unit represents 1 mile. (DOK 3) On Friday, John rode 14 miles farther than he did on Monday. How could his path have changed while remaining rectangular? a) John could have ridden 14 miles farther north. b) John could have ridden 3½ miles farther north and 7 miles farther east. c) John could have ridden 7 miles farther east. d) John could have ridden 1¾ miles farther north and 3½ miles farther east. 5 miles 6 miles Students will need to determine the perimeter of the rectangle graphed to answer this question. Then adding 14 would result in a larger perimeter of 26. P P P Students must remember that each change in a dimension will result in twice as much change in the perimeter. Since Math 6 Notes Geometry: 2-Dimensional and 3-Dimensional Figures Page 22 of 46

23 the perimeter must increase by 14, the only choice that will result in that amount of change would be when John rides an additional 7 miles east. (d) 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: printable nets for the platonic solids, shows figures rotating (cube and tetrahedron only) printable nets, tessellated in full color printable nets for many different solids 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 Math 6 Notes Geometry: 2-Dimensional and 3-Dimensional Figures Page 23 of 46

24 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. 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 Math 6 Notes Geometry: 2-Dimensional and 3-Dimensional Figures Page 24 of 46

25 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: Example: Given the top, side and front views, identify (or draw) the figure. Top View Front View Side View Answer: Math 6 Notes Geometry: 2-Dimensional and 3-Dimensional Figures Page 25 of 46

26 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. For testing at the state level (grade 6) on surface area, only surface area nets made up of from triangles and rectangles will be utilized. Try to have students imagine the process of unfolding The following shows an example of the net of a triangular prism Math 6 Notes Geometry: 2-Dimensional and 3-Dimensional Figures Page 26 of 46

27 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 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 = = 294 The surface area of the prism is 294 cm 2. Math 6 Notes Geometry: 2-Dimensional and 3-Dimensional Figures Page 27 of 46

28 side side 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 = 67 7 = 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 cm into its parts. Label the dimensions. 2 cm Bases Lateral Faces 2 top 15 4 front bottom 15 4 back Find the area of all the surfaces. Bases A bh A 152 A 30 A bh A 152 A 30 A bh A 154 A 60 A bh A 154 A 60 Lateral Faces A bh A 24 A 8 A bh A 24 A 8 Surface Area = Area of the top + bottom + front + back + side + side Surface Area = = 196 The surface area of the prism is 196 cm 2. Math 6 Notes Geometry: 2-Dimensional and 3-Dimensional Figures Page 28 of 46

29 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) = = 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 2 b h 1 A A 18 SA SA The surface area is cm 2. Math 6 Notes Geometry: 2-Dimensional and 3-Dimensional Figures Page 29 of 46

30 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 = X 20 = X 20 = X 20 = 360 The total surface area would be the sum of all the areas 1 A A A 342 SA SA 2484 The surface area is 2484 m 2. Math 6 Notes Geometry: 2-Dimensional and 3-Dimensional Figures Page 30 of 46

31 Example: Find the surface area of the isosceles trapezoidal prism The total surface area would be the sum of all the areas SA SA 143 The surface area is 143 in 2. Example: Find the surface area of the triangular prism. 3 1 A A A The total surface area would be the sum of all the areas SA SA 312 The surface area is 312 cm 2. 1 bh 2 1 A 2 A Math 6 Notes Geometry: 2-Dimensional and 3-Dimensional Figures Page 31 of 46

32 Example: Find the surface area of the rectangular pyramid b h 1 A A a) 206 m 2 b) 312 m 2 c) 302 m 2 d) 216 m 2 1 bh 2 1 A A 48 The total surface area would be the sum of all the areas SA 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: 2-Dimensional and 3-Dimensional Figures Page 32 of 46

33 The net for this solid consists of six rectangles and two oddly shaped bases X 5 = X 15 = 450 Decomposed Area of base = = X 20 = X 25 = X 5 = X 5 = X 15 = X 30 = 1200 The total surface area would be the sum of all the areas SA SA 5010 The surface area is 5010 cm 2. Optional Extension: The formula for Total Surface Area of a rectangular prism is given as: 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: 2-Dimensional and 3-Dimensional Figures Page 33 of 46

34 Notice the absence of the two bases in the following diagram 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 LA LA 4060 OR by taking the Total Surface Area and subtracting the areas of the two bases LA 5010 ( ) 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 ft75 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 600in900 in 3in Math 6 Notes Geometry: 2-Dimensional and 3-Dimensional Figures Page 34 of 46

35 Using those dimensions we can find the volume in cubic inches. The volume is 1,620,000 in 3 which must be converted back into cubic feet. V 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 V V 1,728 cubic inches V 1,728 They must actually divide by 1,728!! That s a BIG difference! 1 ft. 12 in. 1,620, ft 3 of loam will be needed. OR 3 in 1 ft 3 1 ft ft 1 12 in So the new dimensions are 50 ft75 ft in. 4 1 V V The volume is 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 V V V 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 ft 1 yd yd yd 35 yd 3 35 cubic yards of loam will be delivered. Math 6 Notes Geometry: 2-Dimensional and 3-Dimensional Figures Page 35 of 46

36 Sample SBAC Questions Standard: 6.G.1, 6.G.3 DOK: 2 Item Type: TE Difficulty: M Math 6 Notes Geometry: 2-Dimensional and 3-Dimensional Figures Page 36 of 46

37 Math 6 Notes Geometry: 2-Dimensional and 3-Dimensional Figures Page 37 of 46

38 Standard: 6.G.2, 6.NS.3 DOK: 2 Item Type: ER Difficulty: M Math 6 Notes Geometry: 2-Dimensional and 3-Dimensional Figures Page 38 of 46

39 Math 6 Notes Geometry: 2-Dimensional and 3-Dimensional Figures Page 39 of 46

40 Standard: 6.G.4 DOK: 2 Item Type: TE Difficulty: M Math 6 Notes Geometry: 2-Dimensional and 3-Dimensional Figures Page 40 of 46

41 Math 6 Notes Geometry: 2-Dimensional and 3-Dimensional Figures Page 41 of 46

42 Sample Explorations Questions Correct Answer: A Correct Answer: C Correct Answer: B Math 6 Notes Geometry: 2-Dimensional and 3-Dimensional Figures Page 42 of 46

43 Correct Answer: A Correct Answer: A Correct Answer: A Math 6 Notes Geometry: 2-Dimensional and 3-Dimensional Figures Page 43 of 46

44 Correct Answer: A Correct Answer: B Correct Answer: B Math 6 Notes Geometry: 2-Dimensional and 3-Dimensional Figures Page 44 of 46

45 Correct Answer: C Correct Answer: B Math 6 Notes Geometry: 2-Dimensional and 3-Dimensional Figures Page 45 of 46

46 Correct Answer: B Correct Answer: B Math 6 Notes Geometry: 2-Dimensional and 3-Dimensional Figures Page 46 of 46