Circle Theorems Explorer

An inscribed angle is formed by two chords that share an endpoint on the circle. The inscribed angle is always half the central angle that subtends the same arc.

A special case is Thales' Theorem: any inscribed angle in a semicircle (central angle = 180\u00B0) is always exactly 90\u00B0.

A tangent line touches the circle at exactly one point and is always perpendicular (90\u00B0) to the radius at that point. From an external point, the tangent length is

where $d$ is the distance from the external point to the center and $r$ is the radius. Two tangent segments from the same external point are always equal in length.

An arc is a portion of the circle's circumference. A sector is the pie-shaped region bounded by two radii and an arc.

Here $\theta$ is the central angle in radians. In degrees, use $s = \frac{\theta}{360} \times 2\pi r$ and $A = \frac{\theta}{360} \times \pi r^2$.

A chord connects two points on a circle. The perpendicular drawn from the center to any chord bisects (cuts in half) that chord.

Equal chords are equidistant from the center. A chord spanning 180\u00B0 is a diameter, the longest possible chord with length $2r$.