19.1 Cross Sections and Solids of Rotation
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1 Name ass Date 19.1 ross Sections and Soids of Rotation Essentia Question: What toos can you use to visuaize soid figures accuratey? Resource Locker Expore Exporing Nets net is a diagram of the surfaces of a three-dimensiona figure that can be foded to form the three-dimensiona figure. To identify a three-dimensiona figure from a net, ook at the number of faces and the shape of each face. ompete each row of the tabe. Express the circumference of the cyinder as a mutipe of π. Type of Soid Exampe Faces Net rectanguar prism 4 m 3 pairs of congruent rectanges 4 m trianguar prism Sampe: 14 cm 5 cm a pair of congruent trianges and 3 rectanges Sampe: 5 cm 14 cm Modue Lesson 1 DO NOT EDIT--hanges must be made through Fie info orrectionkey=nl-;- ass Date
2 Type of Soid Exampe Faces Net rectanguar pyramid 11 cm 6 cm a rectange and 2 pairs of congruent isoscees trianges 6 cm 11 cm cyinder 7 ft 12 ft 2 congruent circes and a rectange 12 ft 14 ft 14 ft 14π ft 1. Discussion Is there more than one way to draw a net for a soid? re there rues for how the faces of a soid are joined to create a net for it? Yes; E.g., ny arrangement of the faces works as ong as a) each face appears exacty once b) if two faces are joined at an edge in the net, they must be joined at this edge in the soid, and c) the faces form one non-overapping surface. Expain 1 Identifying ross Sections Reca that a cross section is a region of a pane that intersects a soid figure. ross sections of three-dimensiona figures sometimes turn out to be simpe figures such as trianges, rectanges, or circes. Exampe 1 Describe the cross section of each figure. ompare the dimensions of the cross section to those of the figure. The bases of the cyinder are congruent circes. The cross section is formed by a pane that is parae to the bases of the cyinder. ny cross section of a cyinder made by a pane parae to the bases wi have the same shape as the bases. Therefore, the cross section is a circe with the same radius or diameter as the bases. Modue Lesson 1
3 The atera surface of the cone curves in the horizonta direction, but not the vertica direction. Therefore the two sides of the cross section aong this surface are straight ine segments with equa engths. The third side is a diameter of the base of the cone. Therefore, the cross section is a (n) isoscees triange. Its base is the diameter of the cone and its eg ength is the sant height of the cone. 2. pane intersects a sphere. Make a conjecture about the resuting cross section. It wi be a circe. Your Turn Describe each cross section of each figure. ompare the dimensions of the cross section to those of the figure. 3. The bases of the cyinder are parae, so the cross section is a quadriatera with at east one pair of opposite sides parae. The bases of the cyinder meet the atera (curved) surface at right anges, so the cross section must contain four right anges. Therefore, the cross section is a rectange. Its base is smaer than the diameter of the cyinder, because the pane does not pass through the widest part of the cyinder, and its height is the same as the cyinder s. Modue Lesson 1
4 Expain 2 Generating Three-Dimensiona Figures You can generate a three-dimensiona figure by rotating a two-dimensiona figure around an appropriate axis. Exampe 2 Describe and then sketch the figure that is generated by each rotation in three-dimensiona space. right triange rotated around a ine containing one of its egs _ Leg is perpendicuar to, so vertex traces out a circe as it rotates about, and therefore _ traces out a circuar base. The hypotenuse, _, traces out the curving surface of the cone whose base is formed by _. The figure formed by the rotation is a cone. rectange rotated around a ine containing one of its sides D D Sides and are both perpendicuar to, so they each trace _ out congruent circes with common radius D D =. Side is parae to, so it traces out a surface formed by moving a ine at a constant distance from. The figure formed by the rotation is a cyinder. Modue Lesson 1
5 4. Discussion What principes can you identify for generating a soid by rotation of a twodimensiona figure? E.g., ny vertex traces a circe, but ony a side perpendicuar to the axis of rotation (and with one endpoint on the axis) generates a circuar surface. side parae to the axis generates a cyindrica surface. side neither parae nor perpendicuar to the axis generates a conica surface (truncated if neither endpoint is on the axis). Your Turn Describe and then sketch the figure that is generated by each rotation in threedimensiona space. 5. trapezoid with two adjacent acute anges rotated around a ine containing the side adjacent to these anges D Sides and D are neither perpendicuar nor parae to, so they form conica surfaces. Side D is parae to, so it forms a curved cyindrica surface. The figure formed by the rotation is a composite of two cones with a cyinder in between. 6. semicirce rotated around a ine containing its diameter Each point on the semicirce traces out a circe. The figure formed by the rotation is a sphere. Modue Lesson 1
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