Circles are a fundamental geometric shape that frequently appears in GMAT Quantitative Reasoning and Data Sufficiency questions. Understanding their properties, formulas, and how they interact with other shapes is crucial for success. This module will guide you through the essential concepts of circles, from basic definitions to more complex applications.

A circle is defined as the set of all points in a plane that are at a fixed distance from a central point. This fixed distance is known as the radius, and the central point is the center.

Two primary formulas are essential for working with circles: the formula for circumference and the formula for area.

Beyond the full circle, GMAT questions may involve parts of a circle:

The equation of a circle in the Cartesian coordinate system is derived from the distance formula. If a circle has its center at (h, k) and a radius r, its equation is: (x - h)² + (y - k)² = r².

In Data Sufficiency questions involving circles, you'll often be given information about radii, diameters, areas, circumferences, points on the circle, or relationships with other geometric figures. Your goal is to determine if the given statements are sufficient to uniquely determine the value of what's being asked (e.g., the radius, the area, the coordinates of the center).

Key things to look for in DS questions:

• Is the radius or diameter directly given or calculable?
• Is the area or circumference given or calculable?
• Are there enough points on the circle to define it uniquely (typically 3 non-collinear points, or center and one point)?
• Are there relationships with other shapes (tangents, inscribed/circumscribed figures) that provide constraints?

GMAT questions can combine circles with other shapes like squares, triangles, or other circles. Understanding tangency (where a line touches a circle at exactly one point) and intersections is vital. A tangent line is always perpendicular to the radius drawn to the point of tangency.