Solids of a Revolution
A solid of a revolution is a three-dimensional figure obtained by rotating a two-dimensional figure around a straight line or axis that lies in the same plane.
An example of a two dimensional figure that is rotated around a straight line is a revolving door. While the revolving door itself is not really a three-dimensional figure or solid, if this were to be a solid, it would be a circular solid or a cylinder.
Below is an example of a two dimensional figure, that started as a 6x8 unit rectangle graphed on a coordinate plane (diagram 1), that was revolved or rotated around the y-axis (diagram 2), to become a circular solid or cylinder (diagram 3).
Below is an example of a two dimensional figure, that started as a 6x8 unit rectangle graphed on a coordinate plane (diagram 1), that was revolved or rotated around the y-axis (diagram 2), to become a circular solid or cylinder (diagram 3).
A solid of a revolution is a three-dimensional figure obtained by rotating a two-dimensional figure around a straight line or axis that lies in the same plane.
An example of a two dimensional figure that is rotated around a straight line is a revolving door. While the revolving door itself is not really a three-dimensional figure or solid, if this were to be a solid, it would be a circular solid or a cylinder.
Below is an example of a two dimensional figure, that started as a 6x8 unit rectangle graphed on a coordinate plane (diagram 1), that was revolved or rotated around the y-axis (diagram 2), to become a circular solid or cylinder (diagram 3).
Below is an example of a two dimensional figure, that started as a 6x8 unit rectangle graphed on a coordinate plane (diagram 1), that was revolved or rotated around the y-axis (diagram 2), to become a circular solid or cylinder (diagram 3).
Let’s take a look at some two dimensional figures that we can rotate into a solid. We can use this website called Shodor 3D Transmographer to help us rotate them and visualize them.
Example 1:
Let’s rotate this triangle around the \(y\)-axis and see what solid 3D figure we get.
In the website on the right under New Polygon, How Many Vertices? We can enter: 3 for our triangle.
Let’s enter \((0, 4)\), \((0, 8)\) and \((4, 8)\) for our vertices of the triangle. If the view isn’t straight on, click on “Reset View” to reset it.
In the middle area below the graph, under Reflect or Revolve, click on “The line \(x = 0\) is the \(y\)-axis across \(x = 0\)” in order to revolve this triangle around the \(y\)-axis. Then click “Revolve.”
What do you see?
Solution:
This is what you should see. It’s an upside down cone with a height of \(4\) units and a radius of \(8\) units.
Let’s rotate this triangle around the \(y\)-axis and see what solid 3D figure we get.
In the website on the right under New Polygon, How Many Vertices? We can enter: 3 for our triangle.
Let’s enter \((0, 4)\), \((0, 8)\) and \((4, 8)\) for our vertices of the triangle. If the view isn’t straight on, click on “Reset View” to reset it.
In the middle area below the graph, under Reflect or Revolve, click on “The line \(x = 0\) is the \(y\)-axis across \(x = 0\)” in order to revolve this triangle around the \(y\)-axis. Then click “Revolve.”
What do you see?
Solution:
This is what you should see. It’s an upside down cone with a height of \(4\) units and a radius of \(8\) units.
Example 2:
Let’s look a figure that is not right up against the axes. Let’s rotate this rectangle around the \(y\)-axis and see what solid 3D figure we get.
In the website on the right under New Polygon, How Many Vertices? We can enter: \(4\) for a rectangle.
Let’s enter \((-8, -5)\), \((-8, 7)\), \((-2, 7)\) and \((-2, -5)\) for our vertices of the rectangle. If the view isn’t straight on, click on “Reset View” to reset it.
In the middle area below the graph, under Reflect or Revolve, click on “The line \(x = 0\) is the \(y\)-axis across \(x = 0\)” in order to revolve this triangle around the \(y\)-axis. Then click “Revolve.”
What do you see?
Solution:
This is what you should see. It’s a cylinder with an opening of a cylinder inside of it to account for the empty space between the rectangle and the y-axis. The cylinder with have a height of \(12\) units and a radius of \(5\) units if you count from the outside of the cylinder to the \(y\)-axis.
Let’s look a figure that is not right up against the axes. Let’s rotate this rectangle around the \(y\)-axis and see what solid 3D figure we get.
In the website on the right under New Polygon, How Many Vertices? We can enter: \(4\) for a rectangle.
Let’s enter \((-8, -5)\), \((-8, 7)\), \((-2, 7)\) and \((-2, -5)\) for our vertices of the rectangle. If the view isn’t straight on, click on “Reset View” to reset it.
In the middle area below the graph, under Reflect or Revolve, click on “The line \(x = 0\) is the \(y\)-axis across \(x = 0\)” in order to revolve this triangle around the \(y\)-axis. Then click “Revolve.”
What do you see?
Solution:
This is what you should see. It’s a cylinder with an opening of a cylinder inside of it to account for the empty space between the rectangle and the y-axis. The cylinder with have a height of \(12\) units and a radius of \(5\) units if you count from the outside of the cylinder to the \(y\)-axis.
Example 3:
Now let’s look at figure that will rotate around the \(x\)-axis. This is a trapezoid but when rotated will actually will be three different 3D figures put together: a cone, cylinder and another cone.
By now, let’s see if you can figure out what to enter into the website.
Solution:
Hopefully, you see this. It’s a cone with a cylinder and another cone at the end. The height of the first cone is \(8\) units, the height of the cylinder is \(8\) units as well and the height of the \(2\)nd cone is \(6\) units. The radius of all three figures is \(6\) units.
Now let’s look at figure that will rotate around the \(x\)-axis. This is a trapezoid but when rotated will actually will be three different 3D figures put together: a cone, cylinder and another cone.
By now, let’s see if you can figure out what to enter into the website.
Solution:
Hopefully, you see this. It’s a cone with a cylinder and another cone at the end. The height of the first cone is \(8\) units, the height of the cylinder is \(8\) units as well and the height of the \(2\)nd cone is \(6\) units. The radius of all three figures is \(6\) units.
Cross Sections
When calculating the volume of a solid, we can cut the solid open and look at its cross section. A cross section is a \(2\) dimensional figure made by the intersection of a 3D figure and a plane. The shape of the cross section formed by the intersection of a plane and a three-dimensional figure depends on the angle of the plane. For instance, what would the intersection of a plane and a sphere look like?
Let’s take a look at some of these solids when we cut them open in a video.
When calculating the volume of a solid, we can cut the solid open and look at its cross section. A cross section is a \(2\) dimensional figure made by the intersection of a 3D figure and a plane. The shape of the cross section formed by the intersection of a plane and a three-dimensional figure depends on the angle of the plane. For instance, what would the intersection of a plane and a sphere look like?
Let’s take a look at some of these solids when we cut them open in a video.
Here are examples of cross sections of a pyramid.
Great Circles
If a plane intersects a sphere, the two dimensional intersection is either a single point or a circle. If the plane slices through the center of the sphere, then that cross section is a called a great circle. Every great circle divides the sphere into two congruent halves called hemispheres.
If a plane intersects a sphere, the two dimensional intersection is either a single point or a circle. If the plane slices through the center of the sphere, then that cross section is a called a great circle. Every great circle divides the sphere into two congruent halves called hemispheres.
Example 4:
What 2D shapes are created by slicing the 3D figures?
a) Right square pyramid with cross section parallel to base.
What 2D shapes are created by slicing the 3D figures?
a) Right square pyramid with cross section parallel to base.
b) Cube with a cross section through midpoints of three edges.
c) Right cone with a cross section perpendicular to the base and passing through the center.
Solution:
a) Square
a) Square
b) Equilateral triangle
c) Isosceles triangle
Quick Check
1) Rotate this figure around the \(y\)-axis. State what the height and radius of the 3D figure will be.
1) Rotate this figure around the \(y\)-axis. State what the height and radius of the 3D figure will be.
2) Rotate this figure around the \(x\)-axis. State what the height and radius of the 3D figure will be.
3) Rotate this figure around the \(x\)-axis. State what the height and radius of the 3D figure will be.
4) Determine the area of the cross section of the right cone that is perpendicular to the base and passing through the center.