Illustrative Math Alignment: Grade 8 Unit 3

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

This is the Teacher's Guide that accompanies Geometry Applications: 3D Geometry.

Related Resources

To see resources related to this topic click on the Related Resources tab above.

Topic

Geometric Models

Topic

Geometric Models

Topic

Geometric Models

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.

Library of Instructional Resources

To see the complete library of Instructional Resources , click on this link.

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.

Related Resources

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.

Related Resources

This is part of a collection of math examples that focus on volume.

This is part of a collection of math examples that focus on volume.

This is part of a collection of math examples that focus on volume.

This is part of a collection of math examples that focus on volume.

This is part of a collection of math examples that focus on volume.

This is part of a collection of math examples that focus on volume.

This is part of a collection of math examples that focus on volume.

This is part of a collection of math examples that focus on volume.

This is part of a collection of math examples that focus on volume.

This is part of a collection of math examples that focus on volume.

This is part of a collection of math examples that focus on volume.

This is part of a collection of math examples that focus on volume.

This is part of a collection of math examples that focus on volume.

This is part of a collection of math examples that focus on volume.

This is part of a collection of math examples that focus on volume.

This is part of a collection of math examples that focus on volume.

This is part of a collection of math examples that focus on volume.

This is part of a collection of math examples that focus on volume.

This is part of a collection of math examples that focus on volume.

This is part of a collection of math examples that focus on volume.

This is part of a collection of math examples that focus on volume.

This is part of a collection of math examples that focus on volume.

This is part of a collection of math examples that focus on volume.

This is part of a collection of math examples that focus on volume.

This Teacher's Guide provides an overview of the 24 worked-out examples that show how to calculate the surface area of different three-dimensional figures.

Related Resources

To see resources related to this topic click on the Related Resources tab above.

This set of tutorials provides 24 examples of how to find the volume of various 3-dimensional geometric figures.

Related Resources

To see resources related to this topic click on the Related Resources tab above.

This set of tutorials provides 24 examples of how to find the volume of various 3-dimensional geometric figures. NOTE: The download is a PPT file.

6/13/11. In this issue we explore the volcanic eruption in Chile that resulted in a huge plume of smoke and ash that was miles high. We explore the viscosity of lava that makes such eruptions possible.

Related Resources

To see resources related to this topic click on the Related Resources tab above.