|Course:||Michigan Geometry Preview 2012|
|Printed by:||Guest user|
|Date:||Thursday, December 14, 2017, 03:54 PM|
Table of contents
All of the units in the course have focused on two-dimensional shapes such as triangles, quadrilaterals and circles. They are called 2-dimensional because they are a flat surface that has a length and width that define their shape. This unit will focus on three-dimensional shapes. A 3-dimensional shape is a solid that has a length, width and height that define their shape. Examples of 3-D shapes in your everyday life are cubes (dice), cylinders (soup cans), and spheres (balls).
A polyhedron is a three-dimensional figure whose surfaces are polygons. All polyhedra use the same terminology to define their parts.
For a video lesson on identifying the parts of three-dimensional shapes, select the following link:
Right vs. Oblique
There are two kinds of polyhedra, right and oblique. A right polyhedron is a figure whose lateral edge is perpendicular to the base of the figure, so the altitude is the length of the lateral edge. An oblique polyhedron is a figure whose lateral edge is not perpendicular to the base of the figure, therefore the altitude is not a lateral edge of the figure.
An altitude is a perpendicular segment that joins the planes of the bases. The height is the length of the altitude.
A regular polyhedron is a polyhedron whose faces are regular congruent polygons which are assembled the same way around each vertex.
The most common regular polyhedra are known as the platonic solids and are represented below with their corresponding names.
The regular polyhedra known as the star polyhedra are given below with their corresponding names.
Using the figure below, identify the vertices, edges, lateral edges, faces, lateral faces and bases.
Step 1. Identify the vertices.
A, B, C, D, E, F, G, H
Step 2. Identify the edges.
Step 3. Identify the faces.
Step 4. Identify the bases.
On a rectangular prism, there are two sets of bases. The following bases were chosen for this example.
Step 5. Identify lateral faces.
Based on the bases identified, the lateral faces are:
Step 6. Identify the lateral edges.
Based on the bases identified, the lateral edges are:
In a previous unit, you explored the different symmetries of two-dimensional objects: reflectional and rotational. Three-dimensional shapes may also have these symmetries as well.
A three-dimensional shape has reflectional symmetry when there is a plane of symmetry that appears to "cut" the solid into two congruent parts. Reflectional symmetry is also known as mirror symmetry or bilateral symmetry. The figures below show two examples of reflectional symmetry of a cube. There are many more planes of symmetry for this solid.
A three-dimensional shape has rotational symmetry when there is a line through the solid acting as an axis that the solid can rotate around to fit onto itself after a turn of less than 360°. The figure below shows two different axes of symmetry for this solid.
For a video lesson on finding symmetry in three-dimensional shapes, select the following link:
To solidify your understanding of the symmetry of geometric figures, visit the following link to Holt, Rinehart, and Winston Homework Help Online. It provides examples, video tutorials and interactive practice with answers available. The Practice and Problem Solving section has two parts. The first part offers practice with a complete video explanation for the type of problem with just a click of the video icon. The second part offers practice with the solution for each problem only a click of the light bulb away.
The cross-section of an object is the face that would remain if we were to cut cleanly through it. The horizontal cross-section of a right prism or cylinder is the same shape and size as its base. Right prisms and cylinders have the same horizontal cross-section all along their height.
The horizontal cross-section of a pyramid is a similar figure to its base. If a horizontal cut is made toward the top of the pyramid, the cross-section will be smaller than a cross-section toward the bottom of the pyramid, as shown in the picture below.
If the cross-section is oblique, then the shape may or may not be congruent or similar to the base of the solid. In the drawing below, the cone has a horizontal cross section that creates a circle but the oblique cross-section creates an ellipse.
For an interactive on slicing solids to see the resulting cross-section, select the following link:
For a video lesson on finding cross-sections of three-dimensional shapes, select the following link:
A net is a two-dimensional pattern that you can fold to form a three-dimensional figure. They are very useful when finding the surface area of a polyhedron. The drawing below represents the net of the rectangular prism to its right.
If you were to cut out the net and make a fold on each dashed line, it would become the rectangular prism on the right.
For an online lesson with videos about the nets of 3-dimensional figures, select the following link:
For a video lesson on identifying a three-dimensional shape from a net, select the following link:
For an interactive site to determine if a net forms a cube, select the following link:
The best way to understand the connection between the 2-D nets and their 3-D figures is to cut them out and fold them. To visit a website that allows you to print nets that can be cut out, folded and taped together, select the following link:
*Note: These figures could also be used to identify vocabulary terms from this unit like edge, vertex, and face.
You have learned a lot about the features and characteristics of solids. We are now going to describe in greater detail the solids that we will focus on for the rest of this unit: prisms, cylinders, pyramids, cones and spheres. Make sure you can identify all five in the picture below.
A prism is a polyhedron with exactly two congruent, parallel faces called bases. A prism is named by the shape of the bases. A square prism has a square base, a triangular prism has a triangle for its base, a pentagonal prism has a pentagon for its base, etc. The lateral edges in a prism are congruent and parallel.
There are right prisms and oblique prisms. In a right prism, the lateral edges and faces are perpendicular to the bases and the lateral faces are rectangles. The drawing on the left is a right rectangular prism. In an oblique prism the lateral faces are parallelograms. The prism on the right is an oblique triangular prism.
There are an infinite number of polygons. Since prisms are named by their polygonal bases, there are an infinite number of kinds of prisms. The drawings below represent the net of some of the most common prisms.
Cylinders are similar to prisms in that they have two congruent and parallel bases. However, the base of all cylinders are circles - not polygons. Similar to prisms, there are right cylinders and oblique cylinders. The drawing on the left is a right cylinder and the one on the right is an oblique cylinder.
If you "unwrap" the middle section of a cylinder and lay it flat, it is a rectangle. A cylinder's net is made up of a rectangle and two circles (the bases).
Unlike prisms and cylinders, pyramids have only one base. Like prisms, the base is a polygon and the pyramid is named by the shape of its base. For example, a pyramid with a triangle for a base is called a triangular pyramid and a pyramid with a square base is called a square pyramid. The faces of a pyramid are triangles and they meet at a common vertex called the apex. Similar to prisms, there are right pyramids and oblique pyramids. The drawing on the left is a right pyramid and the one on the right is an oblique pyramid.
There are an infinite number of polygons. Since pyramids are named by their polygonal base, there are an infinite number of kinds of pyramids. The drawings below represent the net of some of the most common pyramids.
A cone is similar to a pyramid as it has only one base and comes to a point. The difference between a pyramid and a cone is the base. Cones have circular bases. Like other three-dimensional shapes discussed, there are right cones and oblique cones. The drawing on the left is a right cone and the one on the right is an oblique cone.
The net of a cone is drawn below. It is not as obvious as the previous solids and you may want to explore this one by cutting open a party hat to see that it is indeed a sector of a circle.
A sphere is a three-dimensional shape where all points on its surface are equidistant from the center. The radius of the sphere is the distance from the center to the points on the sphere. Examples of spheres include the earth and basketballs.
The two circles that are represented by dashed lines inside the sphere on the right are called great circles. A great circle is the intersection of a sphere and a plane where the resulting circular cross-section has the same radius as the sphere.
The net of a sphere is represented below.
To solidify your understanding of 3-dimensionsalfigures, visit the following link to Holt, Rinehart, and Winston Homework Help Online. It provides examples, video tutorials and interactive practice with answers available. The Practice and Problem Solving section has two parts. The first part offers practice with a complete video explanation for the type of problem with just a click of the video icon. The second part offers practice with the solution for each problem only a click of the light bulb away.
Sources used in this book:
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Annenberg Foundation 2011, "Cross Sections." http://www.learner.org/courses/learningmath/geometry/session9/part_c/ (accessed 6/13/2011).
Baroody, J. Mr. Baroody's Web Page, "Surface Area of Prisms." http://doversherborn.comcastbiz.net/highschool/academics/math/baroody/GeometryHonors/Class (accessed 6/29/2012).
"Creating Nets." http://www.skwirk.com.au/p-c_s-12_u-238_t-626_c-2321/nsw/maths/3d-space/3d-space/creating-nets (accessed 06/28/11).
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OnlineMathLearning.com, "Geometry-Nets of Solids." Accessed June 29, 2012. http://www.onlinemathlearning.com/geometry-nets.html.
Red Apple Education Ltd., "Cross Sections." Last modified 2012. http://www.skwirk.com.au/p-c_s-12_u-238_t-626_c-2322/nsw/maths/3d-space/3d-space/cross- (accessed 6/29/2012).
Wikipedia, "Regular Polyhedron." Last modified June 18, 2012. Accessed June 29, 2012. http://en.wikipedia.org/wiki/Regular_polyhedron.