Mathematics · Euclidean Geometry

Circle Fundamentals

Reference entry · last updated September 20, 2026

A circle is the set of all points in a plane at a fixed distance, the radius, from a fixed point, the center. The disk is the filled region bounded by the circle. Every subsequent quantity, circumference, arc length, sector area, and chord length, follows from the radius and the central angle.

1. First principles and definitions

Let the center be \(O\) and the radius be \(r\). A point \(P\) lies on the circle exactly when \(OP = r\). The key elements are [1]:

Diameter: a chord through the center, of length \(d = 2r\).
Chord: a segment whose endpoints lie on the circle.
Arc: a connected portion of the circle; a minor arc is less than a semicircle, a major arc more.
Central angle: an angle with its vertex at the center, subtending an arc.
Sector: the region bounded by two radii and an arc.
Tangent: a line meeting the circle at exactly one point.

2. Circumference and pi

The number \(\pi\) is defined as the ratio of a circle's circumference to its diameter, and that ratio is the same for every circle. Therefore [1, 2]:

\( C = \pi d = 2\pi r \)

\(\pi\) is irrational, so no exact finite decimal or fraction represents it. The value \(3.14159\ldots\) is an approximation; the digits never terminate or repeat.

3. Arc length and radians

A full turn is \(360^\circ\) or \(2\pi\) radians. A central angle of \(\theta\) radians subtends an arc whose length is the angle times the radius:

\( s = r\theta \qquad (\theta \text{ in radians}) \)

In degrees the same arc length is a fraction of the circumference:

\( s = \frac{\theta}{360^\circ} \cdot 2\pi r \)

Radians are dimensionless, which is why \(s = r\theta\) needs no conversion factor. Degrees require the explicit \(2\pi/360\) ratio.

4. Sector and segment area

A sector is a fraction of the disk set by its central angle. With the angle measured in radians, its area is [1]:

\( A_{\text{sector}} = \frac{1}{2} r^2 \theta \qquad (\theta \text{ in radians}) \)

With the angle measured in degrees, the same area is the corresponding fraction of the disk. The two formulas are not interchangeable as written, because the same symbol \(\theta\) carries a different numerical measure in each:

\( A_{\text{sector}} = \frac{\theta}{360^\circ} \cdot \pi r^2 \qquad (\theta \text{ in degrees}) \)

A circular segment is the region between a chord and its arc. Its area is the sector area minus the triangle formed by the two radii and the chord. With \(\theta\) in radians, that triangle has area \(\tfrac{1}{2} r^2 \sin\theta\), so:

\( A_{\text{segment}} = \frac{1}{2} r^2 (\theta - \sin\theta) \qquad (\theta \text{ in radians}) \)

5. Chord and angle properties

A line from the center perpendicular to a chord bisects that chord. If the perpendicular distance from the center to the chord is \(d\), the chord has length \( 2\sqrt{r^2 - d^2} \).
Inscribed angle theorem: an angle inscribed in a circle is half the central angle that subtends the same arc.
Thales' theorem: an angle inscribed in a semicircle is a right angle. It is the special case of the inscribed angle theorem at central angle \(180^\circ\).
A tangent is perpendicular to the radius drawn to the point of tangency.
The tangent-chord angle equals the inscribed angle in the alternate segment.

References

^Euclid, Elements, Book III (Circle definitions, chord, tangent, and inscribed angle propositions).
^H.S.M. Coxeter, Introduction to Geometry, 2nd ed., John Wiley & Sons, 1969.