17.1 Cross Sections and Solids of Rotation

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1 Name Class Date 17.1 Cross Sections and Solids of Rotation Essential Question: What tools can you use to visualize solid figures accurately? Explore G.10.A Identify the shapes of two-dimensional cross-sections of prisms, pyramids, cylinders, cones, and spheres and identify three-dimensional objects generated by rotations Exploring Nets Resource Locker A net is a diagram of the surfaces of a three-dimensional figure that can be folded to form the three-dimensional figure. To identify a three-dimensional figure from a net, look at the number of faces and the shape of each face. Use paper and pencil, concrete objects, or other manipulatives to explore three-dimensional figures. Complete each row of the table. Express the circumference of the cylinder as a multiple of π. Type of Solid Example Faces Net 4 m 1.5 m 1.5 m 4 m 2 m 2 m 1.5 m 2 m triangular prism a pair of congruent triangles and 3 rectangles Module Lesson 1

2 Type of Solid Example Faces Net a rectangle and 2 pairs of congruent isosceles triangles 6 cm 11 cm cylinder 7 ft 12 ft Reflect 1. Discussion Is there more than one way to draw a net for a solid? Are there rules for how the faces of a solid are joined to create a net for it? Explain 1 Identifying Cross Sections Recall that a cross section is a region of a plane that intersects a solid figure. Cross sections of three-dimensional figures sometimes turn out to be simple figures such as triangles, rectangles, or circles. Example 1 Describe the cross section of each figure. Compare the dimensions of the cross section to those of the figure. The bases of the cylinder are congruent circles. The cross section is formed by a plane that is parallel to the bases of the cylinder. Any cross section of a cylinder made by a plane parallel to the bases will have the same shape as the bases. Therefore, the cross section is a circle with the same radius or diameter as the bases. Module Lesson 1

3 B The lateral surface of the cone curves in the horizontal direction, but not the vertical direction. Therefore the two sides of the cross section along this surface are straight line segments with equal lengths. The third side is a diameter of the base of the cone. Therefore, the cross section is a (n) triangle. Its base is the of the cone and its leg length is the height of the cone. Reflect 2. A plane intersects a sphere. Make a conjecture about the resulting cross section. Your Turn Describe each cross section of each figure. Compare the dimensions of the cross section to those of the figure. 3. Module Lesson 1

4 4. (Hint: Use the fact that the sphere has complete rotational symmetry about any axis (line through its center), and consider an axis perpendicular to the plane.) Explain 2 Generating Three-Dimensional Figures You can generate a three-dimensional figure by rotating a two-dimensional figure around an appropriate axis. Example 2 Describe and then sketch the figure that is generated by each rotation in three-dimensional space. A right triangle rotated around a line containing one of its legs A l B _ Leg BC is perpendicular to l, so vertex C traces out a circle as it rotates about l, and therefore BC _ traces out a circular base. The hypotenuse, AC _, traces out the curving surface of the cone whose base is formed by BC _. The figure formed by the rotation is a cone. C Module Lesson 1

5 B A rectangle rotated around a line containing one of its sides l B C Sides BC and are both A D to l, so they each trace _ out congruent with common BC =. Side is parallel to l, so it traces out a surface formed by moving a at a constant distance from l. The figure formed by the rotation is a. Reflect 5. Discussion What principles can you identify for generating a solid by rotation of a twodimensional figure? Your Turn Describe and then sketch the figure that is generated by each rotation in threedimensional space. 6. A trapezoid with two adjacent acute angles rotated around a line containing the side adjacent to these angles l B C D A Module Lesson 1

6 7. A semicircle rotated around a line containing its diameter l \ Elaborate 8. Discussion If a solid has been generated by rotating a plane figure around an axis, will the solid always have cross sections that are circles? Will it always have cross sections that are not circles? Explain. 9. Essential Question Check-In What tools can you use to visualize solid figures? Explain how each tool is helpful. Module Lesson 1

7 Evaluate: Homework and Practice 1. Which of the figures is not a net for a cube? Explain. a. c. Online Homework Hints and Help Extra Practice b. d. Describe the three-dimensional figure that can be made from the given net Describe the cross section Module Lesson 1

8 Describe the cross section formed by the intersection of a cone and a plane parallel to the base of the cone. 12. Describe the cross section formed by the intersection of a sphere and a plane that passes through the center of the sphere. Use paper and pencil, concrete objects, or other manipulatives to explore three-dimensional figures. Sketch and describe the figure that is generated by each rotation in three-dimensional space. 13. Rotate a semicircle around a line through the endpoints of the semicircle. 14. Rotate an isosceles triangle around the triangle s line of symmetry. 15. Rotate an isosceles right triangle around a line that contains the triangle s hypotenuse 16. Rotate a line segment around a line that is perpendicular to the segment that passes through an endpoint to the segment. Module Lesson 1

9 17. Multiple Response Which of the following shapes could be formed by the intersection of a plane and a cube? Select all that apply. A. Equilateral Triangle B. Scalene Triangle C. Square D. Rectangle E. Circle 18. A student claims that if you dilate the net for a cube using a scale factor of 2, the surface area of the resulting cube is multiplied by 4 and the volume is multiplied by 8. Does this claim make sense? 19. Find the Error A regular hexagonal prism is intersected by a plane as shown. Which cross section is incorrect? Explain. A B Module Lesson 1

10 H.O.T. Focus on Higher Order Thinking 20. Architecture An architect is drawing plans for a building that is a hexagonal prism. Describe how the architect could draw a cutaway of the building that shows a cross section in the shape of a hexagon, and a cross section in the shape of a rectangle. 21. Draw Conclusions Is it possible for a cross section of a cube to be an octagon? Explain. 22. Communicate Mathematical Ideas A cube with sides of length s is intersected by a plane that passes through three of the cube s vertices, forming the cross section shown. What type of triangle is in the cross section? Explain. s s s Image Credits: Ocean/ Corbis Module Lesson 1

11 23. The three triangles all have the same area because each base and each height are congruent. Make and test a conjecture about the volume of the solids generated by rotating these triangles around the base. A B C Module Lesson 1

12 Lesson Performance Task Each year of its life, a tree grows a new ring just under the outside bark. The new ring consists of two parts, light-colored springwood, when the tree grows the fastest, and a darker-colored summerwood, when growth slows. When conditions are good and there is lots of sun and rain, the new ring is thicker than the rings formed when there is drought or excessive cold. At the center of the tree is a dark circle called pith that is not connected to the age of the tree. Summerwood Springwood Pith Bark 1. Describe the history of the tree in the diagram. 2. The redwood trees of coastal California are the tallest living things on earth. One redwood is 350 feet tall, 20 feet in diameter at its base, and around 2000 years old. Assume that the lower 50 feet of the tree form a cylinder 20 feet in diameter and that all of the rings grew at the same rate. a. What is the total volume of wood in the 50-foot section? Use 3.14 for π. b. How wide is each annual ring (springwood and summerwood combined)? Disregard the bark and pith in your calculations. Show your work. Write your answer in inches. Module Lesson 1