Circles
A circle has no corners and no straight sides. What can you still measure on it, and what does its roundness force to be true?
Where you are 15 inside
What this area is about
A circle is every point in a plane at one distance r from a center. That single condition fixes its size, the lengths and areas of its pieces, and a surprising number of angle and length facts about lines that cut or touch it.
How these fit together
Parts and names. Radius and diameter, chords, secants and tangents and arcs and central angles name what can be drawn on a circle.
Size. Circumference defines \pi and gives C = 2\pi r; area gives A = \pi r^2. Arc length takes a fraction of the circumference, and sectors and segments a fraction of the area (a segment is a sector minus a triangle).
Angle results. The inscribed angle theorem and angles made by chords, secants and tangents tie angles to arcs. Tangent properties start from the right angle a tangent makes with the radius at its point of contact. A quadrilateral is cyclic, its corners on one circle, exactly when its opposite angles add to 180^\circ.
Length results. Chord theorems and segment lengths in circles.
Coordinates. The equation of a circle and finding the center and radius.
A path through it
Radius and diameter first, then chords and circumference. Circumference comes before area and arcs; arcs before arc length and the inscribed angle theorem; area, arcs and chords before sectors. Tangent properties follow chords. After the inscribed angle theorem and tangent properties come angles made by chords, secants and tangents, and the chord theorems, then segment lengths. The equation comes before finding the center and radius. Cyclic quadrilaterals have no fixed place.
Next: proofs about circles and radian measure.