From how a circle relates to a line, to sectors, tangent lines, and inscribed polygons — change the values with sliders and see for yourself why each circle property holds. (The inscribed angle theorem is covered on a separate page.)
A circle is a shape built from a single definition: "the set of points that are the same distance from a center." But this one simple rule naturally gives rise to a surprising range of properties — its relationship with a line, sectors, tangent lines, and inscribed polygons.
For example, just by comparing how far a line sits from the circle's center (distance d) against the radius (r), you can pin down exactly how many times that line meets the circle. An arc's length and a sector's area are also both precisely proportional to the central angle, so knowing the angle alone lets you calculate them right away.
Move the sliders to change the distance, angle, and number of sides, and watch with your own eyes exactly when and how each property changes.
Move the line closer to the circle. Comparing the distance from the center to the line (d) against the radius (r) determines the number of intersection points: 2 (secant) if d<r, 1 (tangent) if d=r, and 0 if d>r.
The larger the central angle, the more the arc length and sector area grow — at exactly the same rate. Compare "what share of the whole circle" with the bar graphs.
Draw tangent lines to a circle from a point outside it, and you always get two — and remarkably, the two tangent lines are always exactly the same length. This holds no matter where you move point P.
Archimedes drew a regular polygon inside a circle and kept increasing its number of sides to calculate pi. See for yourself how the polygon gets closer and closer to the circle as the number of sides (n) increases.