Illustrative Math Alignment: Grade 8 Unit 3

Functions and Volume

Lesson 21: Cylinders, Cones, and Spheres

Use the following Media4Math resources with this Illustrative Math lesson.

Title	Body	Curriculum Topic
Definition--3D Geometry Concepts--Pyramid	Pyramid Topic 3D Geometry Definition A pyramid is a three-dimensional geometric figure with a polygonal base and triangular faces that converge to a single point called the apex. Description In the realm of three-dimensional geometry, a pyramid is a significant shape due to its unique properties and applications. A pyramid consists of a base that can be any polygon, such as a triangle, square, or pentagon, and triangular faces that connect each edge of the base to a single apex point. This structure results in a solid figure that is both aesthetically pleasing and structurally efficient.	Pyramids
Definition--3D Geometry Concepts--Rectangular Prism	Rectangular Prism Topic 3D Geometry Definition A rectangular prism is a three-dimensional figure with six rectangular faces, where opposite faces are congruent and parallel. Description The rectangular prism is a fundamental shape in three-dimensional geometry, serving as a building block for understanding more complex 3D structures. It is characterized by its three dimensions: length, width, and height, which are clearly labeled in the image. This shape is ubiquitous in both natural and man-made environments, making it a crucial concept for students to grasp.	Rectangular Prisms
Definition--3D Geometry Concepts--Slant Height	Slant Height Topic 3D Geometry Definition Slant height is the distance measured along a lateral face from the base to the apex of a three-dimensional figure, such as a pyramid or a cone. Description In the context of three-dimensional geometry, the slant height is a crucial measurement for various solid figures, particularly right pyramids and right circular cones. It represents the shortest path along the surface of the figure from the apex (top point) to the base, distinguishing it from the vertical height which measures the perpendicular distance from the apex to the center of the base.	3-Dimensional Figures
Definition--3D Geometry Concepts--Sphere	Sphere Topic 3D Geometry Definition A sphere is a perfectly round three-dimensional geometric object in which every point on the surface is equidistant from the center. Description In the realm of three-dimensional geometry, a sphere is a fundamental shape characterized by its symmetry and uniformity. It is defined mathematically as the set of all points in space that are at a constant distance, known as the radius, from a fixed point called the center. This distance is the same in all directions, making the sphere a unique object with no edges or vertices.	Spheres
Definition--3D Geometry Concepts--Square Pyramid	Square Pyramid Topic 3D Geometry Definition A square pyramid is a three-dimensional geometric figure with a square base and four triangular faces that converge at a single point called the apex.	Pyramids
Definition--3D Geometry Concepts--Surface Area	Surface Area of 3D Figures Topic 3D Geometry Definition Surface area is the total area that the surface of a three-dimensional object occupies. Description In the realm of three-dimensional geometry, surface area is a fundamental concept that quantifies the extent of a 3D shape's exterior surface. This measure is crucial for various applications, including engineering, architecture, and everyday tasks. For example, when painting a room, the surface area of the walls, ceiling, and floor must be calculated to determine the amount of paint required.	Surface Area
Definition--3D Geometry Concepts--Triangular Prism	Triangular Prism Topic 3D Geometry Definition A triangular prism is a three-dimensional geometric solid with two congruent triangular bases and three rectangular faces. Description The triangular prism is a fundamental shape in three-dimensional geometry, playing a crucial role in understanding the properties of polyhedra and their applications in various fields. This prism is characterized by its unique structure, consisting of two parallel triangular bases connected by three rectangular faces. The shape of the triangular bases can vary, allowing for right, equilateral, isosceles, or scalene triangular prisms.	Triangular Prisms
Definition--3D Geometry Concepts--Triangular Pyramid	Triangular Pyramid Topic 3D Geometry Definition A triangular pyramid, also known as a tetrahedron, is a three-dimensional geometric figure with four triangular faces, six edges, and four vertices. Description In the realm of three-dimensional geometry, the triangular pyramid holds significant relevance due to its unique properties and structural simplicity. Each triangular face of the pyramid converges at a single point known as the apex, forming a solid figure that is both symmetrical and aesthetically pleasing. This geometric shape is the simplest form of a pyramid and is often used in various fields such as architecture, molecular chemistry, and computer graphics.	Pyramids
Definition--3D Geometry Concepts--Vertex	Vertex in 3D Geometry Topic 3D Geometry Definition A vertex is a point where three or more edges meet in a three-dimensional figure. Description In the study of three-dimensional geometry, the term vertex is fundamental. A vertex is a critical point in any 3D geometric shape, marking the intersection of edges. For example, in a cube, each corner where the edges converge is a vertex. Vertices are essential in defining the shape and structure of 3D figures, as they help in understanding the spatial relationships between different parts of the figure.	3-Dimensional Figures
Definition--3D Geometry Concepts--Vertical Cross-Sections of a Cone	Vertical Cross Sections of a Cone Topic 3D Geometry Definition A vertical cross section of a cone is the intersection of the cone with a plane that passes through its vertex and base, resulting in a two-dimensional shape.	Cones
Definition--3D Geometry Concepts--Vertical Cross-Sections of a Cylinder	Vertical Cross Sections of a Cylinder Topic 3D Geometry Definition A vertical cross-section of a cylinder is the intersection of the cylinder with a plane that is parallel to its axis. This cross-section is typically a rectangle if the plane cuts through the entire height of the cylinder.	Cylinders
Definition--3D Geometry Concepts--Vertical Cross-Sections of a Square Pyramid	Vertical Cross Sections of a Square Pyramid Topic 3D Geometry Definition A vertical cross section of a square pyramid is the intersection of the pyramid with a vertical plane that passes through its apex and base, resulting in a two-dimensional shape.	Pyramids
Definition--3D Geometry Concepts--Vertical Cross-Sections of a Triangular Prism	Vertical Cross Sections of a Triangular Prism Topic 3D Geometry Definition A vertical cross section of a triangular prism is a two-dimensional shape obtained by slicing the prism parallel to its height, revealing a triangular face.	Triangular Prisms
Definition--3D Geometry Concepts--Volume	Volume Topic 3D Geometry Definition Volume is the measure of the amount of space occupied by a three-dimensional object, expressed in cubic units. Description Volume is a fundamental concept in the study of three-dimensional geometry. It quantifies the capacity of a 3D object, indicating how much space it occupies. This measurement is crucial in various fields, including mathematics, engineering, architecture, and physical sciences.	Volume
Definition--3D Geometry Concepts--Cavalieri's Principle	Cavalieri's Principle Topic 3D Geometry Definition Cavalieri's Principle states that if two solids are contained between two parallel planes, and every plane parallel to these planes intersects both solids in cross-sections of equal area, then the two solids have equal volumes. Description Cavalieri's Principle is a fundamental concept in three-dimensional geometry that provides a method for determining the volume of solids. Named after the Italian mathematician Bonaventura Cavalieri, this principle is particularly useful for comparing the volumes of solids that might not have straightforward geometric shapes.	3-Dimensional Figures
Geometry Applications Teachers Guide: 3D Geometry	Geometry Applications Teachers Guide: 3D Geometry This is the Teacher's Guide that accompanies Geometry Applications: 3D Geometry. This is part of a collection of teacher's guides. To see the complete collection of teacher's guides, click on this link. Note: The download is a PDF file. Related Resources To see resources related to this topic click on the Related Resources tab above.	Applications of 3D Geometry
Google Earth Voyager Story: The Geometry of Sustainable Architecture, Part 2	Google Earth Voyager Story: The Geometry of Sustainable Architecture, Part 2 Topic Geometric Models	Surface Area, Volume and Rational Functions and Equations
Google Earth Voyager Story: The Mathematics of Pyramids, Part 1	Google Earth Voyager Story: The Mathematics of Pyramids, Part 1 Topic Geometric Models	Pyramids
Google Earth Voyager Story: The Mathematics of Pyramids, Part 2	Google Earth Voyager Story: The Mathematics of Pyramids, Part 2 Topic Geometric Models	Pyramids
INSTRUCTIONAL RESOURCE: Math Examples 55	INSTRUCTIONAL RESOURCE: Math Examples--Surface Area This set of tutorials provides an overview of the 24 worked-out examples that show how to calculate the surface area of different three-dimensional figures. This is part of a collection of math examples for a variety of math topics. To see the complete collection of these resources, click on this link. Note: The download is a PPT file. Library of Instructional Resources To see the complete library of Instructional Resources , click on this link.	Surface Area
Interactive Math Game--Memory Game: 3D Figures	Interactive Math Game--Memory Game: 3D Figures Use this math game to review 3D figures. This is a Memory-style game in which students must remember the location of pairs of identical images. This is part of a collection of math games and interactives. To see the complete collection of the games, click on this link. Note: The download is the teacher's guide. Related Resources To see additional resources on this topic, click on the Related Resources tab.	3-Dimensional Figures
Interactive Math Game: Math Riddles--3D Geometry	Interactive Math Game: Math Riddles--3D Geometry In this Math Riddles Game, have your students review vocabulary around the topic of 3D Geometry. The Math Riddles games are useful for practicing: Math Vocabulary, Key Concepts, Critical Thinking. This is part of a collection of math games and interactives. To see the complete collection of the games, click on this link. Note: The download is the teacher's guide. Related Resources To see additional resources on this topic, click on the Related Resources tab.	3-Dimensional Figures
Math Example--Volume Concepts--Calculating Volume: Example 1	Math Example--Volume Concepts--Calculating Volume: Example 1 Topic Volume Description A rectangular prism with dimensions labeled: length = 30, width = 10, and height = 8. The image shows how to find the volume of the prism using the formula for volume of a rectangular prism. This image illustrates Example 1: The caption explains how to calculate the volume of the rectangular prism using the formula V = l * w * h. The given dimensions are substituted into the formula: V = 30 * 10 * 8 = 2400..	Volume
Math Example--Volume Concepts--Calculating Volume: Example 10	Math Example--Volume Concepts--Calculating Volume: Example 10 Topic Volume Description A green cylinder with a general radius y and height x. The radius is marked on the top surface, and the height is marked on the side. This image illustrates Example 10: The task is to find the volume of this cylinder. The volume formula V = πr2h is used, and substituting r = y and h = x, the volume is calculated as V = xy2π.	Volume
Math Example--Volume Concepts--Calculating Volume: Example 11	Math Example--Volume Concepts--Calculating Volume: Example 11 Topic Volume Description A hollow green cylinder with an outer radius of 10 units, an inner radius of 9 units, and a height of 15 units. The radii are marked on the top surface, and the height is marked on the side. This image illustrates Example 11: The task is to find the volume of this hollow cylinder. The volume formula for a hollow cylinder V = πr12h1 - πr22h2 is used. Substituting values, the result is V = 285π.	Volume
Math Example--Volume Concepts--Calculating Volume: Example 12	Math Example--Volume Concepts--Calculating Volume: Example 12 Topic Volume Description A hollow green cylinder with an outer radius y, an inner radius y - 1, and a height x. The radii are marked on the top surface, and the height is marked on the side. This image illustrates Example 12: The task is to find the volume of this hollow cylinder. Using V = π(r12h1 - r22h2), substituting values gives: V = πx(y2 - (y - 1)2= πx(2y - 1).	Volume
Math Example--Volume Concepts--Calculating Volume: Example 13	Math Example--Volume Concepts--Calculating Volume: Example 13 Topic Volume Description A rectangular-based pyramid is shown with dimensions: base length 10, base width 8, and height 30. The image demonstrates how to calculate the volume of this pyramid. This image illustrates Example 13: The caption provides a step-by-step solution for calculating the volume of a pyramid with a rectangular base using the formula V = (1/3) * Area of Base * h. Substituting values: V = (1/3) * 8 * 10 * 30 = 800.	Volume
Math Example--Volume Concepts--Calculating Volume: Example 14	Math Example--Volume Concepts--Calculating Volume: Example 14 Topic Volume Description A general rectangular-based pyramid is shown with variables x, y, and z representing the base dimensions and height. This example shows how to calculate the volume of a pyramid using variables instead of specific numbers. This image illustrates Example 14: The caption explains how to calculate the volume of a pyramid with a rectangular base using the formula V = (1/3) * Area of Base * h, which simplifies to V = (1/3) * x * y * z.	Volume
Math Example--Volume Concepts--Calculating Volume: Example 15	Math Example--Volume Concepts--Calculating Volume: Example 15 Topic Volume	Volume
Math Example--Volume Concepts--Calculating Volume: Example 16	Math Example--Volume Concepts--Calculating Volume: Example 16 Topic Volume Description A truncated rectangular-based pyramid is shown with variables x, y, and z representing dimensions. The smaller virtual pyramid has reduced dimensions by 3 units for both width and length and reduced height by z - 20. The image demonstrates how to calculate the volume in terms of variables. This image illustrates Example 16: The caption explains how to find the volume of a truncated pyramid using variables for both pyramids' dimensions. Formula: V = (1/3) * xy(z + 20) - (1/3) * (y - 3)(x - 3)(z), which simplifies to V = (1/3) * (xyz + 60x + 60y - 180).	Volume
Math Example--Volume Concepts--Calculating Volume: Example 17	Math Example--Volume Concepts--Calculating Volume: Example 17 Topic Volume Description A green sphere with a radius labeled as 3. The image is part of a math example showing how to calculate the volume of a sphere. This image illustrates Example 17: The text describes finding the volume of a sphere. The formula used is V = (4/3) * π * r3, where r = 3. After substituting, the result is V = 36π. Volume is a fundamental concept in geometry that helps students understand the space occupied by three-dimensional objects. In this collection, each example uses various geometric shapes to calculate volume, showcasing real-life applications of volume in different shapes.	Volume
Math Example--Volume Concepts--Calculating Volume: Example 18	Math Example--Volume Concepts--Calculating Volume: Example 18 Topic Volume Description A green sphere with a radius labeled as x. This image is part of a math example showing how to calculate the volume of a sphere using an unknown radius. This image illustrates Example 18: The text explains how to find the volume of a sphere with an unknown radius x. The formula used is V = (4/3) * π * r3, and substituting r = x gives V = (4/3) * x3 * π.	Volume
Math Example--Volume Concepts--Calculating Volume: Example 19	Math Example--Volume Concepts--Calculating Volume: Example 19 Topic Volume Description A green cube with side length labeled as 7. The image illustrates how to calculate the volume of a cube with known side length. This image illustrates Example 19: The text describes finding the volume of a cube. The formula used is V = s3, where s = 7. After substituting, the result is V = 343. Volume is a fundamental concept in geometry that helps students understand the space occupied by three-dimensional objects. In this collection, each example uses various geometric shapes to calculate volume, showcasing real-life applications of volume in different shapes.	Volume
Math Example--Volume Concepts--Calculating Volume: Example 2	Math Example--Volume Concepts--Calculating Volume: Example 2 Topic Volume Description A rectangular prism with dimensions labeled as x, y, and z. The image shows a general example of calculating the volume of a rectangular prism using variables instead of specific numbers. This image illustrates Example 2: The caption describes how to find the volume of a rectangular prism using variables for length (x), width (y), and height (z). The formula is given as V = x * y * z, but no specific values are provided.	Volume
Math Example--Volume Concepts--Calculating Volume: Example 20	Math Example--Volume Concepts--Calculating Volume: Example 20 Topic Volume Description A green cube with side length labeled as x. This image is part of a math example showing how to calculate the volume of a cube using an unknown side length. This image illustrates Example 20: The text explains how to find the volume of a cube with an unknown side length x. The formula used is V = s3, and substituting s = x gives V = x3.	Volume
Math Example--Volume Concepts--Calculating Volume: Example 21	Math Example--Volume Concepts--Calculating Volume: Example 21 Topic Volume Description A hollow cube with an outer edge of 9 and an inner hollow region with an edge of 7. The image shows how to calculate the volume by subtracting the volume of the inner cube from the outer cube. This image illustrates Example 21: Find the volume of a hollow cube. The formula used is V = s13 - s23, where s1 is the outer edge (9) and s2 is the inner edge (7). The solution calculates 9^3 - 7^3 = 386..	Volume
Math Example--Volume Concepts--Calculating Volume: Example 22	Math Example--Volume Concepts--Calculating Volume: Example 22 Topic Volume Description A hollow cube with an outer edge of x and an inner hollow region with an edge of x - 2. The image shows how to calculate the volume by subtracting the volume of the inner cube from the outer cube. This image illustrates Example 22: Find the volume of a hollow cube. The formula used is V = s13 - s23, where s1 = x and s2 = x - 2. Expanding and simplifying gives V = 6x2 - 12x + 8.	Volume
Math Example--Volume Concepts--Calculating Volume: Example 23	Math Example--Volume Concepts--Calculating Volume: Example 23 Topic Volume Description A cone with a height of 12 and a radius of 4. The image shows how to calculate its volume using the cone volume formula (V = 1/3 * π * r2 * h). This image illustrates Example 23: Find the volume of a cone. The formula used is V = (1/3) * π * r2 * h, where r = 4 and h = 12. Substituting these values gives V = (1/3) * π * (42) * 12 = 64π.	Volume
Math Example--Volume Concepts--Calculating Volume: Example 24	Math Example--Volume Concepts--Calculating Volume: Example 24 Topic Volume Description A cone with a height labeled as y and a radius labeled as x. The image shows how to calculate its volume using the cone volume formula (V = 1/3 * π * r2 * h). This image illustrates Example 24: Find the volume of a cone. The formula used is V = (1/3) * π * r2 * h, where r = x and h = y. Substituting these variables gives V = (x^2 * y)/3 * π.	Volume
Math Example--Volume Concepts--Calculating Volume: Example 3	Math Example--Volume Concepts--Calculating Volume: Example 3 Topic Volume Description A hollow rectangular prism with outer dimensions: length = 60, width = 20, and height = 16. The inner hollow part has dimensions: length = 60, width = 18, and height = 14. The image shows how to subtract volumes to find the hollow volume. This image illustrates Example 3: The caption explains how to calculate the volume of a hollow rectangular prism by subtracting the inner volume from the outer volume. V = (60 * 20 * 16) - (60 * 18 * 14) = 4080.	Volume

Illustrative Math Alignment: Grade 8 Unit 3

Functions and Volume

Lesson 21: Cylinders, Cones, and Spheres

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