Lesson 1

Solids of Rotation

  • Let’s rotate two-dimensional shapes to make three-dimensional shapes.

1.1: Which One Doesn’t Belong: Solids

Which one doesn’t belong?


A photo of a wire spiral toy


A photo of a piece of red chalk


A photo of a structure with a conical roof


A photo of a puzzle cube

1.2: Axis of Rotation

Explore the figure you chose or were assigned in the applet, unchecking the other three boxes. Do not move the slider until after you answer the first question.

  1. Do not move the slider. Notice that the Surface command for each figure specifies which axis will be used to rotate the figures. What do you imagine the shape will look like when you rotate the figure using the slider? Describe any interesting features of the shape.
  2. Move the slider. Describe the solid that is traced out as you rotate the shape around its axis. Use the Rotate 3D Graphics tool to see the shape from other perspectives. 
    Icon for Rotate 3 D Graphics Tool. Vertical semi circle with arrow pointing down. Intersecting horizontal semi circle with arrow pointing left.
  3. Predict what solids will be formed by the shapes of other members of your group. Confirm by selecting those shapes and using the sliders to rotate the figures around the appropriate axis. 

  1. Graph \(y=\text-x+3\) from \(x=1\) to \(x=3\).
  2. Sketch the solid of rotation generated by rotating this line using the \(y\)-axis as the axis of rotation.
  3. What figure is made?
  4. The object being rotated here is a line, not a two-dimensional object like in the lesson. How does this affect the result of the rotation? 

1.3: From Three Dimensions to Two

Students in Team A:

Create a new three-dimensional figure using the applet.

  • Choose one of the 4 figures by deselecting the other 3.
  • Change the points of your selected figure to make a new shape.
  • Use the slider to rotate the shape to create a three-dimensional figure.
  • Hide the two-dimensional shape with the check box.

Students in Team B: 

Try to determine the two-dimensional shape used to create the three-dimensional figure.

  • Do not use the slider.
  • Drag and rotate the three-dimensional figure to see it from multiple angles.
  • Draw the axes and the two-dimensional shape that could be rotated to create the figure.
  • Once you have a two-dimensional shape, show it to your partner.

Check the responses by showing the two-dimensional shape with the check box and using the slider. Switch roles with the other team and create a new shape.


Take a coin and spin it on its edge. Give it good speed and make sure it stays vertical. What shape do you see? When you spin the coin, you should see a sphere.

A circle goes to a sphere that is on an vertical axis and spun. The vertical lines are shown on the sphere.

This triangular flag is made of metal. What shape will you see if the pole is spun quickly?

A horitzontal line with a blue triangle at the top of the line.

When the flag rotates with the pole, you should see a cone.

A horizontal line with green triangles at the top as a result of rotating the line.

In both pictures, the object spins around a line called the axis of rotation, and the solid created is called the solid of rotation. If you were to spin the flag using a different axis of rotation, you would see a different solid of rotation. This is what it looks like when the flag is rotated using a different axis of rotation.

Green triangles shown as if they are spinning on the diagonal axis. They resemble a top that is spinning.

A machine called a lathe cuts away at a rotating block of material. For example, a lathe could be used to make a decorative sculpture shaped like the figure on the right from a block of wood. The image on the left shows the cross-sectional shape that would remain after the lathe carved away part of the spinning block of wood. If we rotate this two-dimensional shape using the vertical axis shown, it produces the sculpture shape.

Vertical dashed line of symmetry with a partial drawning on the right. This goes to a full drawing that is symmetrical. It resembles a wavy vase.

Glossary Entries

  • axis of rotation

    A line about which a two-dimensional figure is rotated to produce a three-dimensional figure, called a solid of rotation. The dashed line is the axis of rotation for the solid of rotation formed by rotating the green triangle.

    Green triangles shown as if they are spinning on the diagonal axis. They resemble a top that is spinning.
  • solid of rotation

    A three-dimensional figure formed by rotating a two-dimensional figure using a line called the axis of rotation.

    The axis of rotation is the dashed line. The green triangle is rotated about the axis of rotation line to form a solid of rotation.

    Green triangles shown as if they are spinning on the diagonal axis. They resemble a top that is spinning.