Aim #6: What are general pyramids and cones? CC Geometry H Do Now: Put the images shown below into the groups (A,B,C and D) based on their properties. Group A: General Cylinders Group B: Prisms Group C: Figures that come to a point with a polygonal base Group D: Figures that come to a point with a curved or semi curved region as a base 1 2 3 4 5 6 7 8 General Cone: Let B be a region in a plane E and V be a point not in E. The cone with base B and vertex V is the union of all segments joining V to all V V points in B. E General Cone vertex B B E General Pyramids Figure 1 Figure 2 Right Circular Cone = "Cone" center of circle h r any region for a base for a base and named by its for a base pyramid pyramid pyramid
The vertex of a "right" general cone lies on a perpendicular to the base that passes through the center of the base. Figure 3 Figure 4 rectangular pyramid - not right right rectangular pyramid circular cone - not right General pyramids have one base and lateral faces and edges. Name the base: Name the lateral faces: Name the lateral edges: Name the base edges: Complete: 1. A general cone with a circle as a base is called a cone. 2. A general cone with a polygonal base is called a. 3. A cone has a height that goes from the vertex to the of the base 4. Name the general cone as specifically as possible. A dilation of three dimensional space with center O and scale factor r is defined the same way it is in the plane. The dilation maps O to itself and maps any other point X to the point X' on a ray OX so that OX' = r (OX) C X X' Y Y'
1) In the following triangular pyramid, cross section ΔA'B'C'is parallel to base ABC. a. What can you conclude about AB and A'B'? Why? b. Since AB ll A'B', we can conclude that, in ΔAVB, A'B' splits sides AV and BV. A dilation maps A to A' and B to B'. What point is the center of dilation? What is the scale factor of this dilation? c. The length of A'B' is times the length of AB. d. What is true about the relationship between BC and B'C'?, AC and A'C'? We can conclude that B'C' is times BC and A'C' is times AC. Therefore, ΔABC ~ ΔA'B'C' by. e. The area of ΔA'B'C' = area of ΔABC. f. If the area of ΔABC is 25 mm 2, what is the area of ΔA'B'C'? 2) In the triangular pyramid, ΔA'B'C' is a cross section that is parallel to the base of the pyramid. Altitude VX intersects the base and cross section at X and V X', respectively. a. By what criterion is ΔAVX ~ ΔA'VX' similar? b. If the distance from X to V is 18 mm, the distance from X' to V is 12 mm, and the area of ΔA'B'C' is 28 mm 2, what is the area of ΔABC? A' 12 X' B' C' 18 C A X B 3) The area of the base of a cone is 16 and the height is 10. Find the area of a parallel cross section that is a distance 5 from the vertex.
4) The base of a pyramid is a trapezoid. The trapezoidal bases have lengths of 3 and 5, and the trapezoid s height is 4. Find the area of the parallel slice that is three-fourths of the way from the vertex to the base. General cone cross-section theorem: If two general cones have the same base area and the same height, then cross-sections for the general cones that are the same distance from the vertex have the same area. 5) The following pyramids have equal altitudes and both bases are equal in area. The cross sections are the same distance from the vertices. If BC = and B'C' = and the area of TUVWXYZ is 30 units 2, what is the area of cross section A'B'C'D'? Let's Sum it Up! - Pyramids are general cones with polygonal bases. - The cross-section of a cone parallel to the base is similar to the base. - The area of the similar region should be the area of the original figure times the square of the scale factor.
Name CC Geometry H Date HW #6 1) The diagram below shows a circular cone and a pyramid. The bases of the cones are equal in area and they have equal heights. a. Sketch a cross-section in each cone that is parallel to the base of the cone and 2/3 closer to the vertex than the base plane. b. If the area of the base of the circular cone is 616 units 2, find the exact area of the slice drawn in the pyramid. 2) The base of a pyramid has area 4. A cross-section that lies in a parallel plane that is distance of 2 from the base plane, has an area of 1. Find the height, h, of the pyramid. 3) The following right triangle is rotated about side AB. What is the resulting figure and what are its dimensions?
4) A general hexagonal pyramid has height 10 in. A slice 2 in. above the base has area 16 in 2. Find the area of the base. 5) A cone has base area 36 cm 2. A parallel slice 5 cm from the vertex has area 25 cm 2. Find the height of the cone. 6) A general cone has base area 36 units 2. Find the area of the slice of the cone that is parallel to the base and 2/3 of the way from the vertex to the base. Review: 1) a. Approximate the area of a disk of radius 4 using an inscribed regular hexagon. b. Find the exact area of a disk that has a radius of 4. c. What is the percent of error, to the nearest tenth, of the approximation of the area in part a?