Topics
11. Modeling in Two & Three Dimensions
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United States of AmericaFL
Grade 912

11.03 Solids of revolution

Lesson
Practice
Lesson

Concept summary

A revolution is a full turn around an axis, or rotation around a point. The angular measure of one revolution is 360 \degree.

When a two-dimensional object is rotated about an axis it forms a three dimensional object called a solid of revolution.

A first and fourth quadrant coordinate plane without numbers with a right triangle plotted on the first quadrant. The base of the triangle is along the positive x axis, and its hypotenuse has a negative slope.
A triangle with one leg along the axis of rotation
A solid of revolution on first and fourth quadrant coordinate plane without numbers. The solid of the revolution resembles a cone in a horizontal position with its center along the x axis.
The resulting solid of revolution

Note that the axis of rotation does not always have to pass through the object to be rotated.

A first and second quadrant coordinate plane without numbers with a rectangle plotted on the second quadrant.
A rectangle away from the axis of rotation
A solid of revolution on a first and second quadrant coordinate plane without numbers. The solid of the revolution resembles a cylinder with a cylindrical hole in its center. The center of the figure is along the positive y axis.
The resulting solid of revolution

Worked examples

Example 1

Sketch the solid of revolution produced by rotating the following shape about the y-axis:

A four quadrant coordinate plane without numbers with a rectangle plotted on the second and third quadrant. One of the lengths of the rectangle is along the y axis.

Solution

Reflect the shape over the y-axis.

A four quadrant coordinate plane without numbers with a rectangle plotted on the second and third quadrant, and the same rectangle reflected over the y axis.

Join the end points on the original figure with the corresponding points on the reflection with ovals.

A solid of revolution on a four quadrant coordinate plane. The solid of the revolution resembles a cylinder with its center along the y axis.

Example 2

Identify the shape that, when rotated about the y-axis, would create the following solid:

A solid of revolution on a first and second quadrant coordinate plane. The solid of the revolution resembles a bowl with its center along the y axis.

Approach

The shape is rotated about the y-axis, so if we find the shape that is on the xy-plane. To do this we take the cross-section and then only consider the part on one side of the y-axis.

Solution

Taking the cross-section we get:

A first and second quadrant coordinate plane without numbers. A curved shaped is drawn on the first and second quadrant. The shaped resembles a parabola that opens upward and the vertex is at the origin, and bounded by a horizontal segment on the top.

Taking the LHS of the x-axis:

A first and second quadrant coordinate plane without numbers. A curved shaped is drawn on the second quadrant. The shape resembles the left side of a parabola that opens upward and the vertex is at the origin, bounded by a horizontal segment on top, and bounded by a vertical segment along the y axis.

Outcomes

MA.912.GR.4.2

Identify three-dimensional objects generated by rotations of two-dimensional figures.

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