Pie menus benefit from Fitts' Law by minimizing the target distance to a small constant (the radius of the inactive region in the menu center where the cursor starts) and maximizing the target area of each item (a wedge shaped slice that extends to the edge of the screen).

They also have the advantage that you don't need to focus your visual attention on hitting the target (which linear menus require), because you can move in any direction into a big slice without looking at the screen (while parking the cursor in a little rectangle requires visual feedback), and you can learn to use them with muscle memory, with quick "mouse ahead" gestures.

http://www.donhopkins.com/drupal/node/100

An Empirical Comparison of Pie vs. Linear Menus

Jack Callahan, Don Hopkins, Mark Weiser (+) and Ben Shneiderman.
Computer Science Department University of Maryland College Park, Maryland 20742
(+) Computer Science Laboratory, Xerox PARC, Palo Alto, Calif. 94303.
Presented at ACM CHI'88 Conference, Washington DC, 1988.

Abstract

Menus are largely formatted in a linear fashion listing items from the top to bottom of the screen or window. Pull down menus are a common example of this format. Bitmapped computer displays, however, allow greater freedom in the placement, font, and general presentation of menus. A pie menu is a format where the items are placed along the circumference of a circle at equal radial distances from the center. Pie menus gain over traditional linear menus by reducing target seek time, lowering error rates by fixing the distance factor and increasing the target size in Fitts's Law, minimizing the drift distance after target selection, and are, in general, subjectively equivalent to the linear style.

http://www.donhopkins.com/drupal/node/98

The Design and Implementation of Pie Menus -- Dr. Dobb's Journal, Dec. 1991

There're Fast, Easy, and Self-Revealing.

Copyright (C) 1991 by Don Hopkins.

Originally published in Dr. Dobb's Journal, Dec. 1991, lead cover story, user interface issue.

Introduction

Although the computer screen is two-dimensional, today most users of windowing environments control their systems with a one-dimensional list of choices -- the standard pull-down or drop-down menus such as those found on Microsoft Windows, Presentation Manager, or the Macintosh.

This article describes an alternative user-interface technique I call "pie" menus, which is two-dimensional, circular, and in many ways easier to use and faster than conventional linear menus. Pie menus also work well with alternative pointing devices such as those found in stylus or pen-based systems. I developed pie menus at the University of Maryland in 1986 and have been studying and improving them over the last five years.

During that time, pie menus have been implemented by myself and my colleagues on four different platforms: X10 with the uwm window manager, SunView, NeWS with the Lite Toolkit, and OpenWindows with the NeWS Toolkit. Fellow researchers have conducted both comparison tests between pie menus and linear menus, and also tests with different kinds of pointing devices, including mice, pens, and trackballs.

Included with this article are relevant code excerpts from the most recent NeWS implementation, written in Sun's object-oriented PostScript dialect.

https://www.youtube.com/watch?v=Jvi98wVUmQA

Demo of Pie Menus in SimCity for X11. Ported to Unix and demonstrated by Don Hopkins.

https://www.youtube.com/watch?v=SG0FAKkaisg

Pet Rock Remote Control: Pie menu remote control touch screen interface for sending commands to pet rocks.

https://www.youtube.com/watch?v=2KfeHNIXYUc

MediaGraph Music Navigation with Pie Menus Prototype developed for Will Wright's Stupid Fun Club: This is a demo of a user interface research prototype that I developed for Will Wright at the Stupid Fun Club. It includes pie menus, an editable map of music interconnected with roads, and cellular automata.

https://www.youtube.com/watch?v=-exdu4ETscs

The Sims, Pie Menus, Edith Editing, and SimAntics Visual Programming Demo:
This is a demonstration of the pie menus, architectural editing tools, and Edith visual programming tools that I developed for The Sims with Will Wright at Maxis and Electronic Arts.