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Three-Dimensional Geometry Spatial Relations Many jobs in the real-world deal with using three-dimensional figures on two-dimensional surfaces. A good example of this is architects use drawings to show what the exteriors of buildings will look like. Three-dimensional figures have faces, edges, and vertices. A face - is a flat surface, and edge - is where two faces meet, and a vertex is where three or more edges meet. Volume is measured in cubic units. See the example below. Isometric dot paper can be used to draw three-dimensional figures. How many faces do most threedimensional figures have? With your isometric dot paper, sketch the drawing below. Make your box 3 units wide, 2 units high, and 5 units long. Now try to sketch the box. After you have sketched the box, try other figures like a cube or pyramid. Drawing threedimensional figures uses a technique called perspective. Here you make a two-dimensional figure look like it is three-dimensional. Let’s try to see if we can draw some threedimensional figures of our own. You will need some isometric dot paper to sketch you drawing. Next, we are going to make a three-dimensional figure using lock blocks and then draw our figure and determine how many blocks are used to make the figure. 3-dimensional objects can also be depicted as 2-dimensional drawings taken at different views. These representations are called orthogonal drawings. The 3-dimensional drawing at the left is represented by the 2-dimensional drawings from the top, front and right-side views. Volume of Prisms and Cylinders Measured in cubic units3 Volumes of Prisms and Cylinders A prism is a threedimensional figure named for the shape of its bases. Triangular prism has triangles for bases. Rectangular prism has rectangles for bases. If all six faces of a prism are squares, it is a cube. Triangular prism In this triangular prism the two bases are triangles. The formula for volume of a triangular prism is V = Bh, where B is area of the base and h is height. Here is another view of a triangular prism. The view on the left shows you how the prism looks in a 3-dimensional view. The view on the right is the base of the prism. V = Bh B = area of the base = area of a triangle V = ½ bh · h V = (.5)(16)(12) = 96 in2 V = Bh height = 12 in V = 96 · 12 V = 1152 in3 Volume of the prism is 1152 in3. Volume is measured in cubic units. Find the volume of the prism Rectangular prism In this rectangular prism the two bases are rectangles. The volume formula is V = Bh V = (lw)h length · width · height Find the volume of the prism V = Bh or V = lwh V = 12 · 8 · 3 V = 288 in3 The volume of the prism is 288 in3. Volume is measured in cubic units. CUBE Here is a 3-dimensional view of a cube. The view on the left is the cube. The view on the right shows the base of the cube. The formula for the volume of a cube: V = Bh V = lwh V = Bh or V =lwh V = 5 · 5 · 5 or 53 V = 125 units3 The volume of the cube is 125 units3. Volume is measured in cubic units. Find the volume of the cube A die is a cube molded from hard plastic. The edge of a typical die measure 0.62 inches. Dice are usually produced in a mold which holds 100 die at a time. To the nearest cubic inch, how much plastic is needed to fill this large mold? When working with word problems, be sure to read carefully to determine what the question wants you to find. This question clearly indicates that you are to compute the volume by stating “to the nearest cubic inch.” Volume of one die = lwh = (.62)(.62)(.62) = 0.238 cubic inches For 100 dice = 23.8 = 24 cubic inches Cylinder: a cylinder is a three-dimensional figure with two circular bases. The volume of a cylinder is the area of the base B times the height h. V = Bh or V = (πr²)h Find the volume of the cylinder V = Bh or V = πr2h V = (π · 42) · 10 V = 502.4 cm3 The volume of the cylinder is 502.4 cm3. Volume is measured in cubic units. Effects of Changing Dimensions By changing the dimensions of a figure, it can have an effect on the volume in different ways, depending on which dimension you change. Lets look at what happens when you change the dimensions of a prism and a cylinder. A juice box measures 3“ by 2“ by 4“. Explain whether doubling the length, width, or height of the box would double the amount of juice the box holds. Original Double length Double width Double height V = lwh V = 3·2·4 V = 24 cu.in. V = lwh V = 6·2·4 V = 48 cu.in V = lwh V = 3·4·4 V = 48 cu.in V = lwh V = 3·2·8 V = 48 cu.in. A juice can has a radius of 1.5 in. and a height of 5 in.. Explain whether doubling the height of the can would have the same effect on the volume as doubling the radius Original Double radius Double height V= V= V= V= V= V= V= V= V= πr²h π·1.5²·5 11.25π cu.in. πr²h π·3²·5 45π cu.in. πr²h π·1.5²·10 22.5π cu.in. Volumes of Pyramids and Cones 1/3 of prisms and cylinders A pyramid is named for the shape of its base. The base is a polygon, and all the other faces are triangles. A cone has a circular base. The height of a pyramid or cone is a perpendicular line measured from the highest point to the base. A cone has a circular base. The height of a pyramid or cone is perpendicular line measured from the highest point to the base. In the cone to the left the height is h and the radius of the circular base is r. The s is the slant height which is used to measure surface area of a cone or pyramid. The volume formula for a cone is V = 1/3Bh or V = 1/3πr²h A pyramid is named for its base. The base is a polygon, and all the other faces are triangles that meet at a common vertex. The height is a perpendicular line from the base to the highest point. The volume formula for a pyramid is V = 1/3Bh V = 1/3(lw)h The volumes of cones and pyramids are related to the volumes of cylinders and prisms. V = πr²h V = Bh V = 1/3πr²h V = 1/3Bh A cone is 1/3 the size of a cylinder with the same base and height. Also, a pyramid is 1/3 the size of a prism with the same height and base. Finding Volumes A practical application Find the volume of the cylinder to the nearest tenth. V = Bh V = πr2 · h V = 3.14 · 32 · 8.6 V = 243.036 cm3 V = 243 cm3 Find the volume of the prism to the nearest tenth V = Bh V=6·8·2 V = 96 cm3 Find the volume of the triangular prism V = Bh V = ½bh · h V = ½(12 · 16) · 12 V = ½(192) · 12 V = ½(2304) V = 1152 in3 Surface Area of Prisms and Cylinders Back to areas2 again Surface area of objects are used to advertise, inform, create art, and many other things. On the left is an anamorphic image, which is a distorted picture that becomes recognizable when reflected onto a cylindrical mirror. One of the most recognizable forms of advertising that uses surface area of an object is the cereal box. If you find the volume, you will find the amount of cereal the box will hold. If you find the surface area of the box you determine how much cardboard is needed to make the box. When you flatten-out a three-dimensional object the diagram is called a net. Which of the following answers is the correct net for the cube. Choose a, b, c, or d. Finding surface area of figures, for example the box below, can be relatively simple. All is needed is to visualize the faces and then use the appropriate area formulas for rectangles and circles. Surface area is the sum of areas of all surfaces of a figure. The figure to the left is a rectangular prism. Notice how many surfaces there are. Lateral surfaces of a prism are rectangles that connect the bases. Top and bottom Left and right Front and back Surface area - is the sum of the areas of all surfaces of a figure. Lateral surfaces - of a cylinder is the curved surface. Surface Area: is the number of square units needed to cover all surfaces of a threedimensional figure. Surface area is the sum of the areas of all surfaces of a figure. The lateral surfaces of a triangular prism are triangles and rectangles. Two triangular bases and three rectangles. Finding Surface Areas Unfolding the figure Find the surface area of the figure SA = (top & bottom) + ( front & back) + (left & right) = 2(8 · 6) + 2(8 · 2) + 2(6 · 2) = 96 + 32 + 24 SA = 152 cm2 Find the surface area of the figure SA = 2(πr2) + lw = 2(area of circle) + (circumference · height) = 2(3.14 · 3.12) + (π6.2) · 12 = 60.3508 + 233.616 = 293.9668 in2 Find the surface area of the figure SA = 2(area of triangle) + (lw) + (lw) + (lw) = 2(½ · 12 · 16) + (20 · 12) + (16 · 12) + (12 · 12) = 192 + 240 + 192 + 144 = 768 in2 New Year’s Eve ball dropped in New York city each year. The ball is made of 2,668 Waterford crystals with 32,256 LED’s that produce about 16 million different colors. So the next time you see an unusual shape, just remember geometry is all around us. US Pavilion at the 1967 World Expo in Montreal, Canada.