DOCSLIB.ORG

Determine the Shape of Each Cross Section Formed by the Intersection Of

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

11-1 Cross Sections and Solids of Revolution Determine the shape of each cross section 3. plane at an angle relative to the base through opposite formed by the intersection of the described faces plane with the solid. 1. plane perpendicular to the base SOLUTION: A plane at an angle relative to the base through opposite faces will intersect 4 faces, any two SOLUTION: adjacent of which are perpendicular to each other. So the cross section is a rectangle. A plane perpendicular to the bases will intersect both bases on a straight line and the curved edge on two ANSWER: parallel lines. The resulting cross section is a rectangle rectangle. Describe the three-dimensional solid generated ANSWER: by rotating each two-dimensional shape around rectangle the given axis. 4. rectangle 2. plane parallel to the base SOLUTION: Rotating a rectangle around a line that is along one of its sides will yield a cylinder. ANSWER: SOLUTION: cylinder A plane parallel to the base of a triangular prism will intersect a cross section that is the same shape as its 5. circle bases. So the cross section is a triangle. ANSWER: triangle SOLUTION: Rotating a circle around a line which it does not intersect will create a shape like a donut, which in math is called a torus. ANSWER: torus (donut) eSolutions Manual - Powered by Cognero Page 1 11-1 Cross Sections and Solids of Revolution 6. triangle 8. plane at an angle relative to the base that intersects the base SOLUTION: Rotating a right triangle around a line along one of its legs will create a right cone, where one leg is the radius and the other leg is the height of the cone. SOLUTION: A plane at an angle relative to the base that intersects ANSWER: the base of a rectangular pyramid will make a cross cone section that has four sides. The cross section will have one pair of parallel sides (the side along the Determine the shape of each cross section base of the pyramid and the opposite side along one formed by the intersection of the described triangular face) and one pair of sides that are not plane with the solid. parallel along two opposite triangular faces. So, the 7. plane at an angle relative to the bases that does not cross section is a trapezoid. intersect either base ANSWER: trapezoid 9. angled plane that intersects the sphere SOLUTION: A plane at an angle relative to the bases that does not intersect either base will intersect only the lateral face of the cylinder. Since it is not parallel to either SOLUTION: base, and does not intersect either base, the shape of Any plane that intersects a sphere in more than one the cross section must be an ellipse. point will intersect the sphere with a cross section that is a circle. ANSWER: ellipse ANSWER: circle eSolutions Manual - Powered by Cognero Page 2 11-1 Cross Sections and Solids of Revolution Describe the three-dimensional solid generated that the slices form each shape. by rotating each two-dimensional shape around a. rectangle the given axis. b. triangle c. trapezoid 10. SOLUTION: Rotating a two dimensional arc around a line will create a hemisphere with a hemisphere cut out. This creates a bowl shape. SOLUTION: a. The cheese slice is in the shape of a triangular ANSWER: prism. The front, right or left view of the cheese slice bowl shape is a rectangle. So, to get a rectangular shape for the slice, one should cut it vertically. 11. rectangle SOLUTION: A rectangle that is rotated around a line parallel to one of its sides that does not intersect the rectangle b. The top view of the cheese slice is a triangle. So, will create a cylinder with a cylinder cut out of its to get a triangular shape for the slice, one should cut center. This could be called an open cylinder, or a it horizontally. tube. ANSWER: open cylinder (tube) 12. exponential function c. The right or left view of the cheese slice is a rectangle. So, to get a trapezoidal shape for the slice, SOLUTION: one should cut it at an angle. The figure that is rotated around the line has two radii which create circles at each end, and between them there is a curve which is not a straight line, so this is not the frustum of a cone, but a shape with a lateral face that curves like a bell or the end of a horn like a trumpet or trombone. ANSWER: horn shape 13. FOOD Describe how the cheese can be sliced so ANSWER: eSolutions Manual - Powered by Cognero Page 3 11-1 Cross Sections and Solids of Revolution a. slice vertically 16. If a plane intersects with a cube at a vertex of the b. slice horizontally cube, what is the shape of the cross section? Explain c. slice at an angle your answer. SOLUTION: Describe each cross section. a triangle; Three faces of the cube meet to form the vertex, so the cross section is a two-dimensional figure with three sides. ANSWER: a triangle; Three faces of the cube meet to form the 14. vertex, so the cross section is a two-dimensional figure with three sides. SOLUTION: 17. UFO Tanya has a model of a UFO. Sketch a two- dimensional figure that could be rotated around an axis to produce a three-dimensional solid similar to the model. See the model on page 799. SOLUTION: A two-dimensional figure that could make the UFO A vertical plane will cut the sphere into two parts model by a rotation would look like half of the picture. with a cross section of a circle. Something like the figure below should produce the model. ANSWER: circle 15. SOLUTION: ANSWER: Sample answer: A horizontal plane passing through the vertex will cut the cone into two parts with a cross section of a triangle. ANSWER: triangle eSolutions Manual - Powered by Cognero Page 4 11-1 Cross Sections and Solids of Revolution 18. DESIGN Describe how you could create a tube with SOLUTION: a length of 10 inches, a diameter of 2 inches, and a a. To get a circle, you want to make a cut parallel to thickness of inch by rotating a 2-D figure around the bases of the cylinder. an axis. Make a sketch and label it. SOLUTION: Take a rectangle 10 inches long and one-quarter inch wide and rotate it around a horizontal axis, with the outer edge of the rectangle at a distance of 1 inch from the axis. ANSWER: b. To get a longer rectangle, you want to make a cut Take a rectangle 10 inches long and one-quarter inch along the length of the cylinder, through the center. wide and rotate it around a horizontal axis, with the outer edge of the rectangle at a distance of 1 inch from the axis. c. To get an oval, you want to make a cut along an angle to the cylinder, but don't cut through either 19. POTTERY A potter creates three-dimensional base. objects by shaping the clay as it spins on a potter’s wheel. Describe the line or curve that could be rotated around a vertical axis to produce the vase shown on page 799. SOLUTION: Sample answer: The curve would have the same shape as the edge of the vase. The curve would look like the letter S stretched vertically. ANSWER: Sample answer: The curve would have the same shape as the edge of the vase. The curve would look like the letter S stretched vertically. 20. COOKIES Michelle is making cookies with a d. To get a shorter rectangle, you want to make a cylindrical roll of cookie dough. Describe how she cut along the length of the cylinder, but don't cut can cut the cookie dough to make each shape. through the center. a. circle b. longest rectangle c. oval d. shorter rectangle eSolutions Manual - Powered by Cognero Page 5 11-1 Cross Sections and Solids of Revolution atoms are arranged in regular geometrical patterns. Sketch a cross section from a horizontal slice of each crystal. Then describe the rotational symmetry about the vertical axis. a. tetragonal ANSWER: a. Make a vertical cut. b. Make a horizontal through the center of the bases. c. Make a diagonal cut not through the bases. d. Make a horizontal cut not through the center of the b. hexagonal bases. 21. ART A piece of clay in the shape of a rectangular prism is cut in half as shown at the right. a. Describe the shape of the cross section. b. Describe how the clay could be cut to make the c. monoclinic cross section a triangle. SOLUTION: SOLUTION: a. a. The cross section from a horizontal slice will look just like the top view of the figure, which appears to be a square. The front view of the prism is a rectangle, so when it is cut vertically, the cross section will be a rectangle. Like all squares, a 90° rotation will produce an image b. identical to the preimage. The crystal appears the same for every 90° rotation about the axis. b. The cross section from a horizontal slice will look just like the top view of the figure, which appears to be a regular hexagon. Three edges meet at each vertex. So, if you cut off a corner of the clay, you get a triangular cross section. ANSWER: a. rectangle Like all regular hexagons, a 60° rotation will produce b.
Recommended publications
  • The Mathematics of Fels Sculptures

    The Mathematics of Fels Sculptures

    THE MATHEMATICS OF FELS SCULPTURES DAVID FELS AND ANGELO B. MINGARELLI Abstract. We give a purely mathematical interpretation and construction of sculptures rendered by one of the authors, known herein as Fels sculptures. We also show that the mathematical framework underlying Ferguson's sculpture, The Ariadne Torus, may be considered a special case of the more general constructions presented here. More general discussions are also presented about the creation of such sculptures whether they be virtual or in higher dimensional space. Introduction Sculptors manifest ideas as material objects. We use a system wherein an idea is symbolized as a solid, governed by a set of rules, such that the sculpture is the expressed material result of applying rules to symbols. The underlying set of rules, being mathematical in nature, may thus lead to enormous abstraction and although sculptures are generally thought of as three dimensional objects, they can be created in four and higher dimensional (unseen) spaces with various projections leading to new and pleasant three dimensional sculptures. The interplay of mathematics and the arts has, of course, a very long and old history and we cannot begin to elaborate on this matter here. Recently however, the problem of creating mathematical programs for the construction of ribbed sculptures by Charles Perry was considered in [3]. For an insightful paper on topological tori leading to abstract art see also [5]. Spiral and twirling sculptures were analysed and constructed in [1]. This paper deals with a technique for sculpting works mostly based on wood (but not necessarily restricted to it) using abstract ideas based on twirls and tori, though again, not limited to them.
  • Chapter 11. Three Dimensional Analytic Geometry and Vectors

    Chapter 11. Three Dimensional Analytic Geometry and Vectors

    Chapter 11. Three dimensional analytic geometry and vectors. Section 11.5 Quadric surfaces. Curves in R2 : x2 y2 ellipse + =1 a2 b2 x2 y2 hyperbola − =1 a2 b2 parabola y = ax2 or x = by2 A quadric surface is the graph of a second degree equation in three variables. The most general such equation is Ax2 + By2 + Cz2 + Dxy + Exz + F yz + Gx + Hy + Iz + J =0, where A, B, C, ..., J are constants. By translation and rotation the equation can be brought into one of two standard forms Ax2 + By2 + Cz2 + J =0 or Ax2 + By2 + Iz =0 In order to sketch the graph of a quadric surface, it is useful to determine the curves of intersection of the surface with planes parallel to the coordinate planes. These curves are called traces of the surface. Ellipsoids The quadric surface with equation x2 y2 z2 + + =1 a2 b2 c2 is called an ellipsoid because all of its traces are ellipses. 2 1 x y 3 2 1 z ±1 ±2 ±3 ±1 ±2 The six intercepts of the ellipsoid are (±a, 0, 0), (0, ±b, 0), and (0, 0, ±c) and the ellipsoid lies in the box |x| ≤ a, |y| ≤ b, |z| ≤ c Since the ellipsoid involves only even powers of x, y, and z, the ellipsoid is symmetric with respect to each coordinate plane. Example 1. Find the traces of the surface 4x2 +9y2 + 36z2 = 36 1 in the planes x = k, y = k, and z = k. Identify the surface and sketch it. Hyperboloids Hyperboloid of one sheet. The quadric surface with equations x2 y2 z2 1.
  • THE GEOMETRY of PYRAMIDS One of the More Interesting Solid

    THE GEOMETRY of PYRAMIDS One of the More Interesting Solid

    THE GEOMETRY OF PYRAMIDS One of the more interesting solid structures which has fascinated individuals for thousands of years going all the way back to the ancient Egyptians is the pyramid. It is a structure in which one takes a closed curve in the x-y plane and connects straight lines between every point on this curve and a fixed point P above the centroid of the curve. Classical pyramids such as the structures at Giza have square bases and lateral sides close in form to equilateral triangles. When the closed curve becomes a circle one obtains a cone and this cone becomes a cylindrical rod when point P is moved to infinity. It is our purpose here to discuss the properties of all N sided pyramids including their volume and surface area using only elementary calculus and geometry. Our starting point will be the following sketch- The base represents a regular N sided polygon with side length ‘a’ . The angle between neighboring radial lines r (shown in red) connecting the polygon vertices with its centroid is θ=2π/N. From this it follows, by the law of cosines, that the length r=a/sqrt[2(1- cos(θ))] . The area of the iscosolis triangle of sides r-a-r is- a a 2 a 2 1 cos( ) A r 2 T 2 4 4 (1 cos( ) From this we have that the area of the N sided polygon and hence the pyramid base will be- 2 2 1 cos( ) Na A N base 2 4 1 cos( ) N 2 It readily follows from this result that a square base N=4 has area Abase=a and a hexagon 2 base N=6 yields Abase= 3sqrt(3)a /2.
  • Sculptures in S3

    Sculptures in S3

    Sculptures in S3 Saul Schleimer and Henry Segerman∗ Abstract We describe the construction of a number of sculptures. Each is based on a geometric design native to the three- sphere: the unit sphere in four-dimensional space. Via stereographic projection, we transfer the design to three- dimensional space. All of the sculptures are then fabricated by the 3D printing service Shapeways. 1 Introduction The three-sphere, denoted S3, is a three-dimensional analog of the ordinary two-dimensional sphere, S2. In n+ general, the n–dimensional sphere is a subset of R 1 as follows: n n+1 2 2 2 S = f(x0;x1;:::;xn) 2 R j x0 + x1 + ··· + xn = 1g: Thus S2 can be seen as the usual unit sphere in R3. Visualising objects in dimensions higher than three is non-trivial. However for S3 we can use stereographic projection to reduce the dimension from four to three. n n n Let N = (0;:::;0;1) be the north pole of S . We define stereographic projection r : S − fNg ! R by x0 x1 xn−1 r(x0;x1;:::;xn) = ; ;:::; : 1 − xn 1 − xn 1 − xn See [1, page 27]. Figure 1a displays the one-dimensional case; this is also a cross-section of the n– dimensional case. For any point (x;y) 2 S1 − fNg draw the straight line L between N and (x;y). Then L meets R1 at a single point; this is r(x;y). N x 1−y (x;y) (a) Stereographic projection from S1 − (b) Two-dimensional stereographic projection applied to the Earth.
  • Lateral and Surface Area of Right Prisms 1 Jorge Is Trying to Wrap a Present That Is in a Box Shaped As a Right Prism

    Lateral and Surface Area of Right Prisms 1 Jorge Is Trying to Wrap a Present That Is in a Box Shaped As a Right Prism

    CHAPTER 11 You will need Lateral and Surface • a ruler A • a calculator Area of Right Prisms c GOAL Calculate lateral area and surface area of right prisms. Learn about the Math A prism is a polyhedron (solid whose faces are polygons) whose bases are congruent and parallel. When trying to identify a right prism, ask yourself if this solid could have right prism been created by placing many congruent sheets of paper on prism that has bases aligned one above the top of each other. If so, this is a right prism. Some examples other and has lateral of right prisms are shown below. faces that are rectangles Triangular prism Rectangular prism The surface area of a right prism can be calculated using the following formula: SA 5 2B 1 hP, where B is the area of the base, h is the height of the prism, and P is the perimeter of the base. The lateral area of a figure is the area of the non-base faces lateral area only. When a prism has its bases facing up and down, the area of the non-base faces of a figure lateral area is the area of the vertical faces. (For a rectangular prism, any pair of opposite faces can be bases.) The lateral area of a right prism can be calculated by multiplying the perimeter of the base by the height of the prism. This is summarized by the formula: LA 5 hP. Copyright © 2009 by Nelson Education Ltd. Reproduction permitted for classrooms 11A Lateral and Surface Area of Right Prisms 1 Jorge is trying to wrap a present that is in a box shaped as a right prism.
  • The Mathematics of Mitering and Its Artful Application

    The Mathematics of Mitering and Its Artful Application

    The Mathematics of Mitering and Its Artful Application Tom Verhoeff Koos Verhoeff Faculty of Mathematics and CS Valkenswaard, Netherlands Eindhoven University of Technology Den Dolech 2 5612 AZ Eindhoven, Netherlands Email: [email protected] Abstract We give a systematic presentation of the mathematics behind the classic miter joint and variants, like the skew miter joint and the (skew) fold joint. The latter is especially useful for connecting strips at an angle. We also address the problems that arise from constructing a closed 3D path from beams by using miter joints all the way round. We illustrate the possibilities with artwork making use of various miter joints. 1 Introduction The miter joint is well-known in the Arts, if only as a way of making fine frames for pictures and paintings. In its everyday application, a common problem with miter joints occurs when cutting a baseboard for walls meeting at an angle other than exactly 90 degrees. However, there is much more to the miter joint than meets the eye. In this paper, we will explore variations and related mathematical challenges, and show some artwork that this provoked. In Section 2 we introduce the problem domain and its terminology. A systematic mathematical treatment is presented in Section 3. Section 4 shows some artwork based on various miter joints. We conclude the paper in Section 5 with some pointers to further work. 2 Problem Domain and Terminology We will now describe how we encountered new problems related to the miter joint. To avoid misunderstand- ings, we first introduce some terminology. Figure 1: Polygon knot with six edges (left) and thickened with circular cylinders (right) 225 2.1 Cylinders, single and double beveling, planar and spatial mitering Let K be a one-dimensional curve in space, having finite length.
  • Cross-Section- Surface Area of a Prism- Surface Area of a Cylinder- Volume of a Prism

    Cross-Section- Surface Area of a Prism- Surface Area of a Cylinder- Volume of a Prism

    S8.6 Volume Things to Learn (Key words, Notation & Formulae) Complete from your notes Radius- Diameter- Surface Area- Volume- Capacity- Prism- Cross-section- Surface area of a prism- Surface area of a cylinder- Volume of a prism- Section 1. Surface area of cuboids: Q1. Work out the surface area of each cuboid shown below: Q2. What is the surface area of a cuboid with the dimensions 4cm, 5cm and 6cm? Section 2. Volume of cuboids: Q1. Calculate the volume of these cuboids: S8.6 Volume Q2. Section 3. Definition of prisms: Label all the shapes and tick the ones that are prisms. Section 4. Surface area of prisms: Q1. Find the surface area of this triangular prism. Q2. Calculate the surface area of this cylinder. S8.6 Volume Q3. Cans are in cylindrical shapes. Each can has a diameter of 5.3 cm and a height of 11.4 cm. How much paper is required to make the label for the 20 cans? Section 5. Volume of a prism: Q1. Find the volume of this L-shaped prism. Q2. Calculate the volume of this prism. Give your answer to 2sf Section 6 . Volume of a cylinder: Q1. Calculate the volume of this cylinder. S8.6 Volume Q2. The diagram shows a piece of wood. The piece of wood is a prism of length 350cm. The cross-section of the prism is a semi-circle with diameter 1.2cm. Calculate the volume of the piece of wood. Give your answer to 3sf. Section 7. Problems involving volume and capacity: Q1. Sam buys a planter shown below.
  • Pentagonal Pyramid

    Pentagonal Pyramid

    Chapter 9 Surfaces and Solids Copyright © Cengage Learning. All rights reserved. Pyramids, Area, and 9.2 Volume Copyright © Cengage Learning. All rights reserved. Pyramids, Area, and Volume The solids (space figures) shown in Figure 9.14 below are pyramids. In Figure 9.14(a), point A is noncoplanar with square base BCDE. In Figure 9.14(b), F is noncoplanar with its base, GHJ. (a) (b) Figure 9.14 3 Pyramids, Area, and Volume In each space pyramid, the noncoplanar point is joined to each vertex as well as each point of the base. A solid pyramid results when the noncoplanar point is joined both to points on the polygon as well as to points in its interior. Point A is known as the vertex or apex of the square pyramid; likewise, point F is the vertex or apex of the triangular pyramid. The pyramid of Figure 9.14(b) has four triangular faces; for this reason, it is called a tetrahedron. 4 Pyramids, Area, and Volume The pyramid in Figure 9.15 is a pentagonal pyramid. It has vertex K, pentagon LMNPQ for its base, and lateral edges and Although K is called the vertex of the pyramid, there are actually six vertices: K, L, M, N, P, and Q. Figure 9.15 The sides of the base and are base edges. 5 Pyramids, Area, and Volume All lateral faces of a pyramid are triangles; KLM is one of the five lateral faces of the pentagonal pyramid. Including base LMNPQ, this pyramid has a total of six faces. The altitude of the pyramid, of length h, is the line segment from the vertex K perpendicular to the plane of the base.
  • Calculus Terminology

    Calculus Terminology

    AP Calculus BC Calculus Terminology Absolute Convergence Asymptote Continued Sum Absolute Maximum Average Rate of Change Continuous Function Absolute Minimum Average Value of a Function Continuously Differentiable Function Absolutely Convergent Axis of Rotation Converge Acceleration Boundary Value Problem Converge Absolutely Alternating Series Bounded Function Converge Conditionally Alternating Series Remainder Bounded Sequence Convergence Tests Alternating Series Test Bounds of Integration Convergent Sequence Analytic Methods Calculus Convergent Series Annulus Cartesian Form Critical Number Antiderivative of a Function Cavalieri’s Principle Critical Point Approximation by Differentials Center of Mass Formula Critical Value Arc Length of a Curve Centroid Curly d Area below a Curve Chain Rule Curve Area between Curves Comparison Test Curve Sketching Area of an Ellipse Concave Cusp Area of a Parabolic Segment Concave Down Cylindrical Shell Method Area under a Curve Concave Up Decreasing Function Area Using Parametric Equations Conditional Convergence Definite Integral Area Using Polar Coordinates Constant Term Definite Integral Rules Degenerate Divergent Series Function Operations Del Operator e Fundamental Theorem of Calculus Deleted Neighborhood Ellipsoid GLB Derivative End Behavior Global Maximum Derivative of a Power Series Essential Discontinuity Global Minimum Derivative Rules Explicit Differentiation Golden Spiral Difference Quotient Explicit Function Graphic Methods Differentiable Exponential Decay Greatest Lower Bound Differential
  • Module-03 / Lecture-01 SOLIDS Introduction- This Chapter Deals with the Orthographic Projections of Three – Dimensional Objects Called Solids

    Module-03 / Lecture-01 SOLIDS Introduction- This Chapter Deals with the Orthographic Projections of Three – Dimensional Objects Called Solids

    1 Module-03 / Lecture-01 SOLIDS Introduction- This chapter deals with the orthographic projections of three – dimensional objects called solids. However, only those solids are considered the shape of which can be defined geometrically and are regular in nature. To understand and remember various solids in this subject properly, those are classified & arranged in to two major groups- Polyhedron- A polyhedron is defined as a solid bounded by planes called faces, which meet in straight lines called edges. Regular Polyhedron- It is polyhedron, having all the faces equal and regular. Tetrahedron- It is a solid, having four equal equilateral triangular faces. Cube- It is a solid, having six equal square faces. Octahedron- It is a solid, having eight equal equilateral triangular faces. Dodecahedron- It is a solid, having twelve equal pentagonal faces. Icosahedrons- It is a solid, having twenty equal equilateral triangular faces. Prism- This is a polyhedron, having two equal and similar regular polygons called its ends or bases, parallel to each other and joined by other faces which are rectangles. The imaginary line joining the centre of the bases is called the axis. A right and regular prism has its axis perpendicular to the base. All the faces are equal rectangles. 1 2 Pyramid- This is a polyhedron, having a regular polygon as a base and a number of triangular faces meeting at a point called the vertex or apex. The imaginary line joining the apex with the centre of the base is known as the axis. A right and regular pyramid has its axis perpendicular to the base which is a regular plane.
  • Section 6.2 Introduction to Conics: Parabolas 103

    Section 6.2 Introduction to Conics: Parabolas 103

    Section 6.2 Introduction to Conics: Parabolas 103 Course Number Section 6.2 Introduction to Conics: Parabolas Instructor Objective: In this lesson you learned how to write the standard form of the equation of a parabola. Date Important Vocabulary Define each term or concept. Directrix A fixed line in the plane from which each point on a parabola is the same distance as the distance from the point to a fixed point in the plane. Focus A fixed point in the plane from which each point on a parabola is the same distance as the distance from the point to a fixed line in the plane. Focal chord A line segment that passes through the focus of a parabola and has endpoints on the parabola. Latus rectum The specific focal chord perpendicular to the axis of a parabola. Tangent A line is tangent to a parabola at a point on the parabola if the line intersects, but does not cross, the parabola at the point. I. Conics (Page 433) What you should learn How to recognize a conic A conic section, or conic, is . the intersection of a plane as the intersection of a and a double-napped cone. plane and a double- napped cone Name the four basic conic sections: circle, ellipse, parabola, and hyperbola. In the formation of the four basic conics, the intersecting plane does not pass through the vertex of the cone. When the plane does pass through the vertex, the resulting figure is a(n) degenerate conic , such as . a point, a line, or a pair of intersecting lines.
  • Lesson 7: General Pyramids and Cones and Their Cross-Sections

    Lesson 7: General Pyramids and Cones and Their Cross-Sections

    NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 7 M3 GEOMETRY Lesson 7: General Pyramids and Cones and Their Cross-Sections Student Outcomes . Students understand the definition of a general pyramid and cone and that their respective cross-sections are similar to the base. Students show that if two cones have the same base area and the same height, then cross-sections for the cones the same distance from the vertex have the same area. Lesson Notes In Lesson 7, students examine the relationship between a cross-section and the base of a general cone. Students understand that pyramids and circular cones are subsets of general cones just as prisms and cylinders are subsets of general cylinders. In order to understand why a cross-section is similar to the base, there is discussion regarding a dilation in three dimensions, explaining why a dilation maps a plane to a parallel plane. Then, a more precise argument is made using triangular pyramids; this parallels what is seen in Lesson 6, where students are presented with the intuitive idea of why bases of general cylinders are congruent to cross-sections using the notion of a translation in three dimensions, followed by a more precise argument using a triangular prism and SSS triangle congruence. Finally, students prove the general cone cross-section theorem, which is used later to understand Cavalieri’s principle. Classwork Opening Exercise (3 minutes) The goals of the Opening Exercise remind students of the parent category of general cylinders, how prisms are a subcategory under general cylinders, and how to draw a parallel to figures that “come to a point” (or, they might initially describe these figures); some of the figures that come to a point have polygonal bases, while some have curved bases.