Nagoya City University 2012: Regular Octahedron

Problem

Find the volume of a regular octahedron with side length $${a}$$, and the radii of the circumscribed sphere and inscribed sphere of this octahedron.

　This is a problem that might be covered in class. Problems asking for the volume of a regular tetrahedron and the radius of its inscribed sphere are common in textbooks and supplementary problem books. This one is a regular octahedron. In either case, it is important to be able to visualize the shape. Based on that, you calculate the necessary lengths.
　Now, how about the following figure? Looking at this figure, can you find the volume and the radius of the inscribed sphere?

You rotate the figure, draw auxiliary lines, and calculate the necessary lengths, but I have created a web application that makes it easy to visualize that process.

The following screen will appear. If the buttons are not displayed, please reload the page.

The two circles on the left are "circular sliders"; you can drag the green dots to rotate the figure around the coordinate axes. In other words, you can rotate the shape. You can think of it as changing your viewpoint. When you press the "Auxiliary Lines" button, red auxiliary lines are drawn as shown in the figure. O is the center of the circumscribed and inscribed spheres. When you press the "Circumscribed Sphere" button, the circumscribed sphere is displayed.

Rendering the spherical surface takes a little time, so if you move the sliders while the sphere is displayed, the movement will be sluggish. Press the button again to hide the sphere.
When you press the "Inscribed Sphere" button, the inscribed sphere is displayed.

Let's change the viewpoint and look for a figure suitable for finding the radius of the inscribed sphere. For example, the following figure. If you understand that the inscribed sphere and the regular octahedron are in contact at the faces, you will arrive at this figure.

※ The figures were created using Cinderella (CindyJS).