The center angle of the circle refers to the angle between the two ends of arc AB in the circle with center OradiusThe ∠ AOB formed by arc AB is called the center angle of the circle.The central angle of the circle is equal to the samearcRightCircumferential angleTwo times of.[1]
The degree of the central angle of a circle is equal to the degree of the arc it is facing.
Relation with arc, chord and chord center distance
In the same circle or equal circle, if one group of quantities of two center angles, two arcs, two chords, and the chord center distance of two chords are equal, then the other groups of quantities are equal.
Understanding: (Definition)
(1) Equal arc equal center angle
(2) When the circumference of the vertex at the center of the circle is equally divided into 360 parts, the center angle of each part is 1 °
(3) Since the arcs opposite the equal central angles in the same circle are equal, the whole circle is also divided into 360 equal parts. At this time, each arc thus obtained is called an arc of 1 °
(4) The degrees of the central angles of the circles are equal to the degrees of their arcs
In the same circle or equal circle, if (1) two center angles, (2) two arcs, (3) two chords, (4) one group of chord center distance on two chords is equal, then their corresponding other groups of quantities are equal respectively
Relation with peripheral angle
In the same circle or equal circle, the same arc or chord is oppositeCircumferential angleIt is equal to half of the central angle of the circle.
Theorem proof: proof.
Make the diameter CD,
∵OA = OB = OC
∴∠OBC = ∠OCB ∠OAC = ∠OCA
∴∠BOD = ∠OBC+∠OCB = 2∠BCD
Namely: ∠ BCD=1/2 ∠ BOD
Similarly: ∠ ACD=1/2 ∠ AOD
∴∠ACB = ∠BCD - ∠ACD
= 1/2（∠BOD - ∠AOD）
= 1/2∠AOB
Calculation formula
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① L (arc length)=(r/180) X π Xn (n is the degree of center angle, the same below);
② S (sector area)=(n/360) X π rtwo；
③ The central angle of the sector circle n=(180L)/(π r) (degrees).
④ K=2Rsin (n/2) K=chord length;N=central angle of the circle to which the chord is aligned, in degrees.
nature
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① The vertex is the center of the circle;
② Both edges intersect the circumference.
③ Nature of center angle: in the same circle or equal circle, the arcs of the same center angle are equal, the chords are equal, and the chord center distance of the chords is equal.In the same circle or equal circle, as long as one pair of four pairs of quantities is equal, the other three pairs must be equal.[2]
④ OnearcThe degree of is equal to the degree of the central angle of the circle it is facing.[3]
⑤ The circumference angle of a semicircle (or diameter) is a right angle;The chord of the 90 ° circumferential angle is the diameter.
