Circle, disk, segment, sector. Formulas, characterizations and properties of circle

1. Formula of the circumference length in terms of the diameter:

C = πD

2. Formula of the circumference length in terms of the radius:

C = 2πr

1. Formula of the circle area in terms of the radius:

A = πr²

2. Formula of the circle area in terms of the diameter:

A = πD²4

1. Tangent is always perpendicular to the radius of the circle drawn at the point of contact.

2. The shortest distance from the center of circle to tangent is radius of the circle.

3. If two tangents from touch points B and C on the same circle are not parallel, they intersect at point A and the point of contact between the segment and the point of intersection of a tangent is the same segment on a different tangent:

AB = AC

Also, if you draw a line through the center O and the intersection point A of these tangents, the angles between this line and tangents will be equal:

∠ОAС = ∠OAB

1. If a point outside the circle (Q) obtained two secant, crossing the circle at two points A and B for a first secant and C and D for another secant, the products of two intersecting segments are equal:

AQ ∙ BQ = CQ ∙ DQ

2. If the point comes out of Q circle secant, crossing the circle at two points A and B, and the tangent point of contact C, then product segments the secant lengths equal to the square of the tangent:

AQ ∙ BQ = CQ²

1. Formula of the chord length in terms of the radius and central angle:

AB = 2r sin α2

2. Formula of the chord length in terms of the radius and inscribed angle:

AB = 2r sin α

1. Two equal chords tightening two identical arc:

if chords AB = CD, then

arc ◡ AB = ◡ CD

2. If the chord are parallel, the arcs between them will be the same:

if chords AB ∣∣ CD, then

◡ AD = ◡ BC

3. If radius of circle is perpendicular to the chord, it divides chord in half at the point of intersection:

if OD ┴ AB, then

AC = BC

4. If two chords AB and DE intersect at point Q, then the product segments, formed at the intersection, one chord is the product of a different chord sections:

AQ ∙ BQ = DQ ∙ QC

5. Chords of equal length are equidistant from the center of circle.

if chords AB = CD, then

ON = OK

6. The greater chord the closer it is to the center.

if CD > AB, then

ON < OK

1. All inscribed angles, based on one arc is equal (one end of the chord).

2. Inscribed angle will be a straight (90°), if it is based on the diameter of the circle.

3. Any inscribed angle is always equal to half the central angle, based on the same arc

β = α2

4. If two inscribed angles based on a chord and located on either side of her, the sum of the angles is 180°.

α + β = 180°