Quadrilaterals, Polygons - Regular Polygons, Circle

Annulus and annulus segment
Circle
A circle is a set of points that are at the fixed distance (called the radius r) from a fixed point called the center O
Central angle, inscribed angle
A central angle is double the inscribed angle (formed when two secant lines intersect on the circle) subtended by the same arc.
Proof:  Angles b1 and b2 are external angles of the isosceles triangles AOC and BOC, hence
b1 = 2a1b2 = 2a2a = a1+ a2    =>    b = b1+ b2 = 2(a1+ a2) = 2a
Inscribed angles subtended by the same arc are equal. An angle inscribed in a semicircle is a right angle.
Construction of a tangent from a point P to a circle c.
 The midpoint of the line segment OP is the circumcenter of the quadrilateral PD1OD2. The lines PD1 and PD2 are tangents from P to the given circle c.
 Circumference, the length of a circle - the perimeter:
Length of an arc:
 From the proportion
Circle and circular sector
By substituting P = 2pr and R = r in the formula for the area of a regular polygon, obtained is the formula for the area of a circle, that is :

 substituting
Circular segment
The portion of a circle bounded by an arc and a chord is called a segment.
Symbols used in the formulas: c -chord, r -radius, h -height of a segment, A -area of a segment.

Annulus and annulus segment
Annulus or ring is the region enclosed between two concentric circles.
Annulus Annulus segment

Geometry and use of trigonometry contents - A