Ready To Go On? Skills Intervention 10-1 Solid Geometry

Transcription

1 10A Find these vocabulary words in Lesson 10-1 and the Multilingual Glossary. Vocabulary Ready To Go On? Skills Intervention 10-1 Solid Geometry face edge vertex prism cylinder pyramid cone cube net cross section Classifying Three-Dimensional Figures Classify the figure. Name the vertices, edges, and bases. What shape are the bases? What shape are the faces? I J vertex K H L edge G Classify the figure. Name the vertices. face Name the edges. D C E B A F Name the bases. Identifying a Three-Dimensional Figure From a Net Describe the three-dimensional figure that can be made from the given net. A. The net has two congruent faces that are. The remaining faces are. The net forms a. B. The net has face that is a pentagon. The remaining faces are. The net forms a. Describing Cross Sections of Three-Dimensional Figures Describe the cross section. How many sides does the cross section of this figure have? What is the shape of the cross section of the figure?. 141 Holt Geometry

2 10A Ready To Go On? Skills Intervention 10-2 Representations of Three-Dimensional Figures Find these vocabulary words in Lesson 10-2 and the Multilingual Glossary. Vocabulary orthographic drawing isometric drawing perspective drawing vanishing point horizon Drawing Orthographic Views of an Object Draw all six orthographic views of the given object. Assume there are no hidden cubes. Orthographic drawings are drawings of the figure from six different sides: top, bottom, front, back, left, and right. To draw orthographic views, pretend you are looking at the figure from that view. Top Right Bottom Back Left Front 142 Holt Geometry

3 10A Find this vocabulary word in Lesson 10-3 and the Multilingual Glossary. Examining Polyhedrons Find the number of vertices, edges, and faces of a pentagonal pyramid. Use your results to verify Euler s Formula. Number of vertices: Number of edges: Number of faces: Ready To Go On? Skills Intervention 10-3 Formulas in Three Dimensions V E F Euler s Formula states that V E F 2. Substitute the numbers into Euler s Formula: 2 Simplify. 2 Does your result verify Euler s Formula? Vocabulary polyhedron Finding Distances and Midpoints in Three Dimensions Find the distance between the given points. Find the midpoint of the segment with the given endpoints. Round to the nearest tenth, if necessary. (5, 18, 12) and ( 4, 12, 14) distance: d ( x 2 x 1 ) 2 ( y 2 y 1 ) 2 ( z 2 z 1 ) 2 Use the Distance Formula Substitute known values Simplify. Simplify. midpoint: M x 1 x 2, y 1 y 2, z 1 z Use the Midpoint Formula. M 5, 2 12, Substitute known values. M 2, 2, 2,, Simplify. 143 Holt Geometry

4 10A Ready To Go On? Problem Solving Intervention 10-3 Formulas in Three Dimensions When you move around, the distance you travel can be modeled on a threedimensional coordinate plane. A hiker stops for lunch at a picnic area 75 yards north and 30 yards east from her starting point. The elevation of the picnic area is 40 yards higher than the starting point. What is the distance from her starting point to the picnic area? Understand the Problem 1. How can you use geometry to solve this problem? 2. What coordinate does each direction represent? Make a Plan 3. What ordered triple represents the hiker s starting point? 4. What ordered triple represents the picnic area? 5. State the Distance Formula. Solve 6. Substitute the ordered triples into the Distance Formula and simplify. d ( 0 ) 2 ( 0 ) 2 (40 ) What is the distance from the hiker s starting point to the picnic area? x z (30, 75, 40) y Look Back 8. Find the length of the diagonal of a 30 yd by 75 yd by 40 yd rectangular prism. 9. Does this verify the answer you got in Exercise 6? Explain. 144 Holt Geometry

5 10A Ready To Go On? Quiz 10-1 Solid Geometry Classify each figure. Name the vertices, edges, and bases. 1. F 2. M 3. E A D B C L H I L M G J K N Describe the three-dimensional figure that can be made from the given net Describe each cross section Holt Geometry

6 10A Ready To Go On? Quiz continued 10-2 Representations of Three-Dimensional Figures Use the figure made of unit cubes for Exercises 10 and 11. Assume there are no hidden cubes. 10. Draw all six orthographic views. 11. Draw an isometric view. 12. Draw a cube in one-point 13. Draw a rectangle in two-point perspective. perspective Formulas in Three Dimensions Find the number of vertices, edges, and faces of each polyhedron. Use your results to verify Euler s formula. 14. a square pyramid 15. a hexagonal prism 16. a triangular prism 17. A bird flies from its nest to a point that is 12 feet east, 9 feet south, and 8 feet higher in the tree than the nest. How far is the bird from the nest? Find the distance between the given points. Find the midpoint of the segment with the given endpoints. Round to the nearest tenth, if necessary. 18. (0, 0, 0) and (9, 30, 50) 19. (7, 5, 15) and (8, 3, 17) 20. ( 1, 3, 1) and ( 5, 7, 1) 146 Holt Geometry

7 10A Ready To Go On? Enrichment Three-Dimensional Figures Answer each question. 1. The cube at the right is built from 64 smaller cubes. The cube is spray painted on all sides. How many of the cubes are painted on three faces? How many cubes are painted on two faces? How many cubes are painted on only one face? 2. A triangle has vertices J(4, 1, 3), K( 2, 6, 7), and L(10, 4, 1). Classify the triangle by the length of its sides. Find the perimeter of the triangle. 3. A polygon is called a Platonic Solid if all of its faces are regular polygons and if each vertex is the point of intersection of the same number of edges. The Platonic Solids are pictured at the right. Complete the table below. Verify that Euler s Formula works for the Platonic Solids. Tetrahedron Name Shape of face Number of edges at each vertex Number of Faces Number of Edges Number of Vertices Tetrahedron Hexahedron Octahedron Octahedron Dodecahedron Dodecahedron Icosahedron Icosahedron Does Euler s Formula work for each Platonic Solid? 147 Holt Geometry

8 Ready To Go On? Skills Intervention 10-4 Surface Area of Prisms and Cylinders Find these vocabulary words in Lesson 10-4 and the Multilingual Glossary. Vocabulary lateral face lateral edge right prism oblique prism altitude surface area lateral surface axis of a cylinder right cylinder oblique cylinder Finding Lateral Areas and Surface Areas of Prisms Find the lateral area and surface area of the rectangular prism. Round to the nearest tenth, if necessary. lateral area: 6 ft L Ph Use the formula for lateral area. P 2( ) 2( ) ft 2 P is the perimeter of the base. L ( )( ) Substitute known values. 4 ft 5 ft L ft 2 Simplify. surface area: S L 2B Use the formula for surface area. B ( )( ) ft 2 B is the area of the base. S 2( ) Substitute known values. S ft 2 Simplify. Finding Lateral Areas and Surface Areas of Right Cylinders Find the lateral area and surface area of the right cylinder. Give your answer in terms of. lateral area: 20 mm 15 mm L 2 r h Use the formula for lateral area. L 2 ( )( ) Substitute known values. L mm 2 Simplify. surface area: S L 2 r 2 Use the formula for surface area. S 2 ( ) 2 Substitute known values. S mm 2 Simplify. 148 Holt Geometry

9 Ready To Go On? Skills Intervention 10-5 Surface Area of Pyramids and Cones Find these vocabulary words in Lesson 10-5 and the Multilingual Glossary. Vocabulary vertex of a pyramid regular pyramid slant height of a regular pyramid altitude of a pyramid vertex of a cone axis of a cone right cone oblique cone slant height of a right cone altitude of a cone Finding Lateral Areas and Surface Areas of Pyramids Find the lateral and surface area of the square pyramid. lateral area: L 1 2 P P 4s 32 in. 2 Use the formula for lateral area. P is the perimeter of the base. L 1 ( )( ) Substitute known values. 2 L in. 2 Simplify. surface area: S L B B ( ) 2 B is the area of the base of the pyramid. S Substitute known values. S in. 2 Simplify. 8 in. 5 in. Finding Lateral Areas and Surface Areas of Right Cones Find the lateral area and surface area of the right cone. lateral area: L r Use the formula for lateral area of a cone. L ( )( ) Substitute known values. L cm 2 Simplify. surface area: S L B Use the formula for surface area of a cone. B r 2 ( ) cm 2 B is the area of the base of the cone. S Substitue known values. S cm 2 Simplify. 15 cm 39 cm 149 Holt Geometry

10 Ready To Go On? Skills Intervention 10-6 Volume of Prisms and Cylinders Find this vocabulary word in Lesson 10-6 and the Multilingual Glossary. Vocabulary volume Finding Volumes of Prisms Find the volume of the prism. Round to the nearest tenth, if necessary. V wh Use the formula for volume of a right rectangular prism. V ( )( )( ) Substitute known values. V 280 ft 3 Simplify. 10 ft Finding Volumes of Cylinders Find the volume of the cylinder. Give your answer both in terms of and rounded to the nearest tenth. V r 2 h Use the formula for volume of a cylinder. V ( ) 2 ( ) Substitute known values. V cm 3 Simplify. Leave your answer in terms of. V cm 3 Round your answer to the nearest tenth. 7 ft 12 cm 4 ft 8 cm Exploring Effects of Changing Dimensions The dimensions of a triangular prism are multiplied by 2. Describe the effect on the volume. Volume using original dimensions: V Bh Use the formula for volume of a prism. B 1 2 ( )( ) in.3 B is the area of the base. V ( )( ) 336 in. 3 Substitute known values and simplify. Volume after multiplying each dimension by 2: V Bh 12 in. 6 in. Use the formula for volume of a prism. B 1 2 ( )( ) in.3 B is the area of the base. 8 in. 14 in. 16 in. 28 in. V ( )( ) 2688 in. 3 Substitute known values and simplify. When the dimensions of a triangular prism are multiplied by 2, the volume is multiplied by or. 150 Holt Geometry

11 Ready To Go On? Problem Solving Intervention 10-6 Volume of Prisms and Cylinders If you know the volume and the density of a three-dimensional object, you can find its weight. A stone fireplace measures 6 m by 4 m by 11 cm. Find the volume of the fireplace. If the density of stone is 2515 kilograms per cubic meter, what is the mass of the fireplace in kilograms? Understand the Problem 1. What two measurements must you calculate? 2. How many meters is 11 centimeters? 3. What conversion factor will you use to calculate the mass of the fireplace? Make a Plan 4. What formula will you use to find the volume of the fireplace? 5. After you find the volume, how will you find the mass? Solve 6. Calculate the volume of the fireplace. 7. Multiply your answer from Exercise 6 by the conversion factor from Exercise What is the mass of the fireplace in kilograms? Look Back 9. Use this formula to check your solution: density mass volume Substitute the values you found for the mass and volume. density mass volume Does your answer check? 151 Holt Geometry

12 Finding Volumes of Pyramids Find the volume of the pyramid. V 1 3 Bh Ready To Go On? Skills Intervention 10-7 Volume of Pyramids and Cones Use the formula for volume of a pyramid. B ( ) 2 49 yd 2 Find B, the area of the base of the pyramid. V 1 ( )( ) Substitute known values. 3 V yd 3 Simplify. 7 yd 9 yd 7 yd Finding Volumes of Cones Find the volume of the cone. Give your answer both in terms of and rounded to the nearest tenth. V 1 3 Bh V 1 3 r 2 h Use the formula for volume of a cone. B is the area of a circle. V 1 3 ( )2 ( ) Substitute known values. 11 in. 6 in. V 3 in. 3 Simplify. Leave your answer in terms of. V in. 3 Round your answer to the nearest tenth. Finding Volumes of Composite Three-Dimensional Figures Find the volume of the composite figure. Round to the nearest tenth. Find the volume of the cube. V s 3 Use the formula for volume of a cube. V ( ) 3 Substitute and simplify. V cm 3 Find the volume of the square pyramid. V 1 3 Bh Use the formula for volume of a pyramid. V 1 3 ( )2 ( ) Substitute and simplify. V 3 cm3 Subtract the volume of the pyramid from the volume of the cube to find the volume of the composite figure. 5 cm 5 cm 5 cm V composite 3 cm Holt Geometry

13 Ready To Go On? Skills Intervention 10-8 Spheres Find these vocabulary words in Lesson 10-8 and the Multilingual Glossary. Vocabulary sphere center of a sphere radius of a sphere hemisphere great circle Finding Volumes of Spheres Find each measurement. Give your answer in terms of. A. the volume of the sphere V 4 3 r 3 Use the formula for volume of a sphere. 15 m V 4 3 ( )3 Substitute 15 for r. V m 3 Simplify. B. the volume of the hemisphere r 1 2 d 1 ( ) Find the radius of the hemisphere. 2 V 2 3 r 3 Use the formula for volume of a hemisphere. V 2 3 ( )3 Substitute known values. 24 yd V yd 3 Simplify. Finding Surface Areas of Spheres Find the surface area of the sphere. Give your answer in terms of. S 4 r 2 Use the formula for surface area of a sphere. S 4 ( ) 2 Substitute known values. S in. 2 Simplify. 8 in. Exploring Effects of Changing Dimensions The radius of the sphere is multiplied by 1. Describe the effect 2 on the volume. original dimensions: radius multiplied by 1 2 : V 4 3 r 3 V 4 3 r 3 10 ft V 4 3 ( ) 3 f t 3 V ( ) 3 f t If the radius is multiplied by 1 2, the volume is multiplied by or. 153 Holt Geometry

14 Ready To Go On? Problem Solving Intervention 10-8 Spheres Changing the dimensions of a three-dimensional object affects the volume of the object. A scale model of the solar system has a sphere with radius 3.4 cm to represent Mars and a sphere with radius 6.4 cm to represent Earth. About how many times as great is the volume of Earth as the volume of Mars? Understand the Problem 1. What does scale model mean? 6.4 cm 2. What measurements must you find? 3.4 cm Make a Plan 3. What formula will you use to calculate the volumes of the spheres? 4. After you find the volumes of the spheres, what will you do next? Solve 5. Find the volume of the sphere representing Earth. 6. Find the volume of the sphere representing Mars. 7. Divide your answer from Exercise 5 by your answer from Exercise About how many times greater is the volume of the Earth? Look Back 9. How many times greater is the radius of the Earth than the radius of Mars? 10. Cube your answer to Exercise Does your answer to Exercise 10 match your answer to Exercise 8? 154 Holt Geometry

15 Ready To Go On? Quiz 10-4 Surface Area of Prisms and Cylinders Find the surface area of each figure. Round to the nearest tenth, if necessary cm 4 in. 9 m 3 cm 3 cm 7 in. 5 in. 5 m 6 cm 4. The dimensions of an 8 yd by 14 yd by 10 yd right rectangular prism are multiplied by 3. Describe the effect on the surface area Surface Area of Pyramids and Cones Find the surface area of each figure. Round to the nearest tenth, if necessary. 5. a regular hexagonal pyramid with base edge length 9 ft and slant height 14.3 ft 6. a right cone with diameter 22 mm and height 13 mm 7. the composite figure formed by two pyramids shown at right 8 ft 8 ft 8 ft 10-6 Volume of Prisms and Cylinders Find the volume of each figure. Round to the nearest tenth, if necessary. 8. a regular pentagonal prism with base area 28 in. 2 and height 7 in. 9. a cylinder with radius 12 yd and height 15 yd 155 Holt Geometry

16 Ready To Go On? Quiz continued 10. A brick patio measures 18 ft by 16 ft by 6 in. Find the volume of the bricks. If the density of the bricks is 130 pounds per cubic foot, what is the weight of the patio in pounds? 11. The dimensions of a hexagonal prism with base area 374 cm 2 and height 18 cm are tripled. Describe the effect on the volume Volume of Pyramids and Cones Find the volume of each figure. Round to the nearest tenth, if necessary m 12 cm 17 cm 14 m 27 m 11 yd 22 yd 10-8 Spheres Find the surface area and volume of each figure. Give your answers in terms of. 15. a sphere with diameter 26 in. 16. a hemisphere with radius 7 m 17. A junior basketball has a diameter of approximately 7 in., and a regulation basketball has a diameter of approximately 9.5 in. About how many times as great is the volume of the regulation basketball as the volume of the junior basketball? 156 Holt Geometry

17 Ready To Go On? Enrichment Surface Area and Volume Answer each question. 1. The volume of a sphere has the same measure in cubic units as does its surface area in square units. What is the radius of the sphere? 2. A backyard swimming pool measures 10 ft by 14 ft and is 5 ft deep. The pool has a leak and is losing water at the rate of 4 in. per day. If the owner does not refill the water, in how many days will the pool be empty? 3. In the figure at the right, if you rotate the rectangle around the dashed line, the resulting figure will be a cylinder with radius 3 cm and height 7 cm. Find the surface area of the cylinder. Leave your answer in terms of. Find the volume of the cylinder. 4. What figure results from rotating the figure at the right around the dashed line? Find the surface area of the figure. Find the volume of the figure. 5. The figure at the right is an octahedron. Each face is an equilateral triangle. The length of each edge is 18 in. Find the surface area of the octahedron. 5 m 3 cm 13 m 7 cm 18 in. Find the volume of the octahedron. 6. Daniel is repainting the walls in his room. His room is 15 ft wide, 12 ft long and 8 ft high. He plans to put two coats of paints on each wall. One gallon of paint covers approximately 400 ft 2. How many gallons of paint must Daniel buy? Explain your answer. 157 Holt Geometry