Aristotle's Wheel Paradox

When using GeoGebra, or any other software, for measuring; one should keep in mind that the truth value from any pattern found, is questionable if one omits the mathematical reasoning.



Using the tool bar

You can find various tools for making measurements in the tool bar.

Note that when hovering the mouse over a tool, you get a message about what objects are needed. When measuring an angle you must either click on three points or two lines/segments.

Using these tools you can measure, angles, areas, distances and slopes.



You can show units on the axes. Click on the "Preferences" wheel-icon in the upper right corner and choose "Graphics".

There is one tab for the x-axis and one for the y-axis.


If you want to show measurements in text boxes using some unit, you can specify a unit yourself.


To change the text that is created when you make a measurement, open the properties window and add a unit at the end of the text.



Using a command

All the tools in the tool bar have corresponding commands that you can enter in the input bar. The commands act on various objects and these objects must be entered as parameters. You write the parameters within square brackets.

Square brackets, i.e "[" and "]", are entered by pressing Alt+ ( or Alt+).

All commands can be shown in a list if you click on the icon to the right of the input bar. If you click on a command, the parameters needed to use the command will be shown.

If you for example want to measure a distance, you write Dis; after doing so GeoGebra suggests Distance[]. You accept the suggestion by clicking Enter, the command is written and the cursor is moved to inside the square brackets where you can fill in the parameters needed, i.e the names of the points.

If you don't know what parameters are needed, press F1. The F1-button means help in most programs.

A variable holding the value of the distance is created and shown in the Algebra View. When using a command instead of a tool, no text box showing the value measured is created. You can measure the area of a circle or angles in a similar way.

Measuring Angles

In the properties window, you can choose to always display angles within a chosen interval.

Angles are named using Greek letters in GeoGebra. When measuring an angle between three points, there are two different angles that are possible depending on whether you click on the points in a counterclockwise direction or not. If you don't get the angle you wanted, you can undo by either typing Cmd/Ctrl+z or by clicking on the undo-icon at the top right corner icon.


When measuring the angle between two lines, you may have to add an extra point on one of the lines to be able to specify the angle you need.

Apart from the regular object properties, such as colour or size, there are some special properties just for angles. You can decorate angles to mark those angles that are equal. The decorations are found under the Decoration tab in the properties window.


animated gifs:

Aristotle's Wheel Paradox on tumblr.

further info:

Broken Centimeter Ruler:

Measuring and Drawing Angles with GeoGebra:

Wolfram MathWorld: Aristotle's Wheel Paradox

by Malin Christersson under a Creative Commons Attribution-Noncommercial-Share Alike 2.5 Sweden License