Sketchpad’s menus are arranged thematically. While the File, Edit, and Display menus contain commands relating to your Sketchpad documents and workflow, the other menus and tools are more mathematical in nature. Each of these menus presents a distinct mathematical viewpoint and commands appropriate to that viewpoint. Familiarizing yourself with their organizational structure can help you plan your approach to a given construction problem or mathematical challenge.

The Compass and Straightedge tools provide the fundamental tools of compass-and-straightedge Euclidean geometry.

In that the Compass tool does not retain a fixed radius, it technically provides a collapsible, rather than a noncollapsible, compass.

This menu contains additional commands for working in compass-and-straightedge Euclidean geometry. (Many of the objects you create with this menu could be created with the Compass and Straightedge tools alone, though some would take many steps to create with those tools.) Where the command is equivalent to the Compass tool, Construct | Circle by Center+Radius allows you to construct circles of a given radius, acting as a noncollapsible, rather than a collapsible, compass.

This menu contains commands drawn from the perspective of a metric transformational geometry. Use them to construct or explore symmetries and other transformational relationships. You may specify transformational parameters — such as angles of rotation or scaling factors of dilation — either geometrically, by referring to existing objects, or metrically, by entering numeric angles and lengths (or by referring to numerical values and calculations already defined in your sketch).

Since compass-and-straightedge geometry does not include tools for specifying lengths metrically, you cannot construct a segment of a given length — say, 5.0 cm — using only the Construct menu. However, since the Transform menu’s operations are metric in nature, you can use its commands to produce such a result.

This menu’s commands continue the metric theme and offer a variety of ways to determine numeric relationships in your construction. The commands  in the top part of this menu can be thought of as ruler-and-protractor operations: They measure distances, areas, and angles using the metric units you choose in Preferences. The commands that appear in the bottom part of the menu are analytic in nature and measure quantities in relationship to some (existing or newly-defined) coordinate system

This menu’s commands allow you to create, calculate, and tabulate numerical values and functions, moving from the realm of number into algebra.

This menu’s commands continue the analytic perspective, offering operations relating to coordinate systems and graphing.

While each of these menus reflects a unique mathematical perspective, in the course of any Sketchpad activity you may move back and forth between perspectives to focus on different aspects of your activity. In particular, move from a geometric or spatial visualization to a numeric perspective using commands from the Measure menu. (Think of these commands as “turning shapes into numbers.”) Move from numbers back into geometric or spatial visualizations (“turn numbers into shapes”) by using the Plot Points and Plot Function commands, or use Transform menu commands with marked numeric values as transformational values.

Finally, in addition to the commands that produce or construct specific mathematical relationships in your sketch, each menu contains one command that produces a generalization of an arbitrary set of such relationships over some change.

The Construct menu’s Locus command lets you visualize the position of a constructed object over a change in one point’s position.

The Transform menu’s Iterate command lets you visualize the orbit of one or more objects over some number of repetitions of a construction.

The Measure menu’s Calculate command lets you express a general relationship arithmetically between two or more measured quantities.

The Number menu’s Tabulate command lets you analyze the values of a set of measurements over time.

The Graph menu’s Plot Function command lets you visualize a general function evaluated over a domain.

The Toolbox’s Custom tool lets you generalize a set of relationships constructed between objects into a new tool that you can use to replicate that construction on a new set of objects.

Mastering these advanced commands allows you to move beyond the specific mathematical relationships, objects, tools, and commands that form Sketchpad’s starting points and opens up a set of mathematical curves, shapes, and construction tools limited only by your imagination.