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3.5 min video

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Volume of Prisms and Cylinders by Shmoop

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Math Antics 8 min video

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Total Surface Area - the trick to getting it right

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Definition of DENSITY with examples

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teach the earth:enhancing your teaching:quantitative skills

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Jump to: Rock and Mineral density | Rock and mineral specific gravity You can download the questions (Acrobat (PDF) 25kB Jul24 09) if you would like to work them on a separate sheet of paper. Calculating densities ...

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Related Topics:
Common Core (Geometry)
Common Core for Mathematics

Videos, examples, solutions, and lessons to help High School students learn how to identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

Common Core: HSG-GMD.B.4

The following figures show some solids of rotation: cone, cylinder, frustum, sphere. Scroll down the page for more examples and solutions.

Geometry Cross sections
Examples:
What is the shape of the cross section of each of the following figures?
Cross Sections
Learn to identify cross sections of solids.
Examples:
Sketch and identify the cross section of each of the figure below.

Cross Sections & Rotations of 2D to 3D
A video that shows how you can convert Figures from 2D to 3D by cross sections and rotations. Solids of rotation (Solids in 3D)
Rotate a triangle to get a cone.
Rotate a rectangle to get a cylinder
Rotate a trapezium to get frustum
Rotate a circle (semi-circle) to get a sphere. Solids of rotation
A solid of rotation is the three-dimensional (solid) object formed by rotating a two-dimensional area around an axis. For animations to explore the shapes of two-dimensional cross-sections of three-dimensional objects and identify three-dimensional objects generated by rotations of two-dimensional objects, see:

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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Cross Sections and Solids of Rotation (solutions, examples, videos, lessons, worksheets, games, activities)

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Related Topics: Common Core (Geometry) Common Core for Mathematics Videos, examples, solutions, and lessons to help High School students learn how to identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Common Core: HSG-GMD.B.4 The following figures show some solids of rotation: cone, cylinder, frustum, sphere.

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In this section, we will be rotating (spinning) a two-dimensional figure about a line to produce a three-dimensional shape, called a solid of revolution.

An example of a rotation about a line is the concept of a revolving door. The panel of one section of the glass door is "pushed" about a center pole, creating an outer cylindrical path around the pole. While the revolving door does not create a "solid", it does illustrate the process of revolution. If the door filled the entire space as it revolved, a cylindrical solid would be formed.

With the idea of revolution (rotation) in mind,
let's start revolving some geometric shapes to create geometric solids.

Note: The 2-D figure may appear on coordinate axes, from which you can obtain measurement information about the figure. Rotations will most likely be about the y-axis or the x-axis (but may be about any vertical or horizontal line.)
(Rotations about slanted lines in the coordinate plane will not be discussed at this level.)

* A rectangle, shown in the coordinate plane (below, left), is to be rotated about the y-axis. Sketch the resulting solid on the grid and find its volume in cubic units.
The 3-D sketch, which is a cylinder, is shown below on the right.

Using the coordinate grid, the width of the rectangle is 3 units and its height is 4 units.

Note: It may be the case that a portion of a function (or functions) will represent the 2-D figure on coordinate axes to be revolved about an axis or a vertical or horizontal line.
(Rotations about slanted lines in the coordinate plane will not be discussed at this level.)

* The function f ( x) = x ², on the domain 0 < x < 1, is graphed. The resulting graph is then revolved about the y-axis. Sketch the 3-D solid.
Will the volume of this solid be greater than or less than a cone of radius 1 unit and a height of 1 unit?

The volume of this shape will be slightly greater than the volume of a cone with comparable dimensions. This figure is broader near the vertex than a straight sided cone.

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Rotations - Solids of Revolution - MathBitsNotebook(Geo - CCSS Math)

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In this section, we will be rotating (spinning) a two-dimensional figure about a line to produce a three-dimensional shape, called a solid of revolution. An example of a rotation about a line is the concept of a revolving door.

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Cavalieri's principle in 3D (video) | Khan Academy

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Cavalieri's principle tells us that if 2 figures have the same height and the same cross-sectional area at every point along that height, they have the same volume.

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Comparing the area of a circle to the area of a parallelogram

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Comparing the area of a circle to the area of a parallelogram

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Geometry - Informal argument for the formula of the area of a circle