Circle — definition, parts, formulas, properties and uses

Definition and basic terms

A circle is the set of all points in a plane that lie at the same distance from a single point called the center. This formal idea is often given as a definition in elementary geometry. The constant distance from the center to any point on the circle is the radius; the segment from the center to the perimeter is commonly denoted r. The point at the middle is the center, often labeled O.

Named parts and relations

Key parts of a circle include the diameter (a segment passing through the center with endpoints on the circle), the chord (a segment with both endpoints on the circle), the arc (a portion of the circumference), and the sector (the region between two radii and the enclosed arc). A diameter is twice the radius, d = 2r. Radius, diameter and chord are basic vocabulary; more advanced relations involve tangents (lines touching the circle at exactly one point) and secants (lines that meet the circle at two points).

Formulas: circumference and area

The circumference (perimeter) of a circle measures the distance around it and is given by C = 2πr, equivalently C = πd. The area of the flat region enclosed by a circle (sometimes called a disk) equals A = πr². These formulas use the constant π, defined as the ratio of a circle's circumference to its diameter. The number π appears in many branches of mathematics because of the circle's role in geometry and periodic phenomena. Circumference and area formulas are central in problems involving circular shapes.

Geometric and analytic properties

Circles are highly symmetric: they have infinite lines of reflective symmetry through the center and rotational symmetry about the center by any angle. In coordinate geometry a circle with center at the origin and radius r satisfies x² + y² = r². Constructions with compass and straightedge can produce many classical circle-related objects, such as regular polygons inscribed in the circle or tangents at given points. The unit circle (r = 1) is especially important in trigonometry because coordinates on that circle correspond to cosine and sine values. Unit circle considerations simplify many calculations.

History and notable facts

Circles have been studied since ancient times for both practical construction and theoretical reasons. Early mathematicians approximated the constant π by comparing perimeters of inscribed and circumscribed polygons; this approach dates to classical Greek and other ancient traditions. Later proofs established that π is irrational and, further, transcendental, which implies certain classical constructions—such as squaring the circle with only compass and straightedge—are impossible. For historical background and deeper reading, see historical notes and further resources.

Applications and important distinctions

Circles appear widely in engineering, design and the natural world: wheels and gears use circular profiles for smooth rotation; lenses and orbits approximate circular shapes; and many mechanical and architectural elements exploit circular symmetry for strength and aesthetics. Practical calculations often require computing arc length, sector area, or coordinates along a circumference. It is useful to distinguish a circle (the one-dimensional curve) from a disk (the two-dimensional area it bounds). Common examples and problems include measuring sectors, packing circles, and inscribing shapes within a circle.

• Every diameter passes through the center: it is the longest possible chord.
• Tangent lines are perpendicular to the radius at the point of tangency.
• Concentric circles share the same center but have different radii.