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TAKS: Measurement & 3-D • 8.7.D: Locate and name points on a coordinate plane using ordered pairs of rational numbers. • 8.8.A: Find surface area of prisms and cylinders using [concrete] models and nets (2-D models). • 8.8.B: Connect models to formulas for volume of prisms, cylinders, pyramids, and cones. • 8.8.C: Estimate answers and use formulas to solve application problems involving surface area and volume. • 8.10.B: Describe the resulting effect on volume when dimensions of a solid are changed proportionally. Points on a Coordinate Plane A coordinate grid is used to locate and name points on a plane. Points on a Coordinate Plane The x-axis and y-axis divide the coordinate plane into four regions, called quadrants. Example Example Polyhedron A solid formed by polygons that enclose a single region of space is called a polyhedron. Separate your Geosolids into 2 groups: Polyhedra and others. Parts of Polyhedrons • Polygonal region = face • Intersection of 2 faces = edge • Intersection of 3+ edges = vertex face edge vertex Example Separate your Geosolid polyhedra into two groups where each of the groups have similar characteristics. These are the only kind of polyhedra on the TAKS test. What are the names of these groups? Prism A polyhedron is a prism iff it has two congruent parallel bases and its lateral faces are parallelograms. Classification of Prisms Prisms are classified by their bases. Right & Oblique Prisms Prisms can be right or oblique. What differentiates the two? Example Pyramid A polyhedron is a pyramid iff it has one base and its lateral faces are triangles with a common vertex. Classification of Pyramids Pyramids are also classified by their bases. Example The three-dimensional figure formed by spinning a two dimensional figure around an axis is called a solid of revolution. Cylinder A cylinder is a 3-D figure with two congruent and parallel circular bases. • Radius = radius of base • Axis = segment connecting centers of bases Cone A cone is a 3-D figure with one circular base and a vertex not on the same plane as the base. • Altitude = perpendicular segment connecting vertex to the plane containing the base (length = height) Sphere A sphere is the set of all points in space at a fixed distance from a given point. • Radius = fixed distance • Center = given point Sections When a solid is cut by a plane, the resulting plane figure is called a section. A section that is parallel to the base is a cross- section. Example Example Nets Imagine cutting a 3-D solid along its edges and laying flat all of its surfaces. This 2-D figure is a net for that 3-D solid. Example Match one of the red, rubbery nets with its corresponding 3-D solid. Which of the shapes has no net? Example There are generally two types of measurements associated with 3-D solids: surface area and volume. Which of these can be easily found using a shape’s net? Surface Area The surface area of a 3-D figure is the sum of the areas of all the faces or surfaces that enclose the solid. • Asking how much surface area a figure has is like asking how much wrapping paper it takes to cover it. Lateral Surface Area The lateral surface area of a 3-D figure is the sum of the areas of all the lateral faces of the solid. • Think of the lateral surface area as the size of a label that you could put on the figure. Example The net can be folded to form a cylinder. What is the approximate total surface area of the cylinder? Example Using the Formula Chart You could just as easily compute the surface area using a formula. Height vs. Slant Height On the formula chart, h represents height and l represents slant height. Height vs. Slant Height On the formula chart, h represents height and l represents slant height. Example Find the total surface area of the square pyramid. Volume Volume is the measure of the amount of space contained in a solid, measured in cubic units. – This is simply the number of unit cubes that can be arranged to completely fill the space within a figure. Example Find the volume of the given figure in cubic units. More Formulas! The volume of a solid is also easily computed with a formula. What does the B represent? Example Example Example Which solid has the greater volume: a cylinder 3 inches high with a radius of 2 inches or a cone of the same radius that is 8.5 inches high? Example A pipe in the shape of a cylinder with a 30-inch diameter is to go through a passageway shaped like a rectangular prism. The passageway is 3 ft high, 4 ft wide, and 6 ft long. The space around the pipe is to be filled with insulating material. Example 10 What is the volume of the insulating material? Example Find the volume of a cube with a side length of 2 inches. Now find the volume of a cube with a side length of 4 inches. How do the volumes compare? Example Find the volume of a cube with a side length of 2 inches. Now find the volume of a cube with a side length of 4 inches. How do the volumes compare? Volumes of Similar Figures If two solids have a scale factor of a:b, then the corresponding volumes have a ratio of a3:b3. Similarity Relationships For two shapes with a scale factor of a:b, each of the following relationships will be true. Example Example Example A breakfast-cereal manufacturer is using a scale factor of 2.5 to increase the size of one of its cereal boxes. If the volume of the original cereal box was 240 in.3, what is the volume of the enlarged box? Assignment 10th TAKS Practice Workbook: • P. 85, 97, 99, 101, 107