In the study of geometry, especially in circles, the concept of angles is crucial. One of the most significant notions in geometry is the concept of angle. Although the ideas of equality, sums, and differences of angles are fundamental and employed throughout geometry, trigonometry is a discipline based on angle measurement. There are various ways to draw an angle in a circle, each with its method of calculating its size. Central, inscribed, interior, and external angles are the four sorts of angles. We examined numerous types of angles in the angles section, but there are generally four types of angles in the case of a circle. An angle created by the radii, chords, or tangents of a circle is called a circle angle. There are four types of angles: central, inscribed, interior, and outside. Let us have a look at each one of the angles separately.

  • Central angle:

A central angle is produced by two radii, with the vertex in the circle’s centre. The vertex of a central angle is located in the centre of the circle, while the sides of the angle are located on two of the circle’s radii. The diameter of the central angle is the same as the diameter of the arc cut out of the circle by the two sides.

  • Calculation of central angle

Suppose we have a circle at centre O and OA and OB are the radii of the circle then the angle formed by both the radii at the centre is the Central angle.

Central angle = (Arc length*360°)/2πr

Where r is the radius of the circle and arc length is the arc of the circle subtended by the point A and B.

  • Inscribed angle:

An inscribed angle is formed by two crossing chords and has its apex “on” the circle. The vertex of an inscribed angle is on the circle, and the sides of the angle are on two of the circle’s chords. The inscribed angle is half the length of the arc formed by the two sides cutting the circle in half.

Inscribed angle = ½ x intercepted arc

Standing on the arc refers to the angle occupied by an arc. The two theorems are renamed as follows:

  • A circle’s circumferential angle is equal to half of the angle in the centre of the same arc.
  • Two circumferential angles standing on the same arc are equal.
  • Interior angle:

The vertex of an interior angle is the point where two lines join inside a circle. The crossing lines form the angles’ sides. The average of the dimensions of the two arcs carved out of the circle by the intersecting lines is the measure of an interior angle.

  • Exterior angle: –

The vertex of an external angle is when two rays share a terminal outside of a circle. Those two rays represent the angle’s sides. The difference between the intercepted arcs’ measures is divided by two to get the measure of an external angle.

Some of the important theorem related to the angle of the circle

  • The chords will be of identical length if two angles subtended at the centre by two chords are equal.
  • At the centre of the circle, two equal chords subtend equal angles.
  • If drawn from the centre of the circle, the perpendicular to a chord bisects the chord.
  • A perpendicular chord is bisected by a straight line running through the centre of a circle.
  • In the cyclic quadrilateral, the sum of opposite angles is 180°.
  • Angles subtended by the same arc at any point on the circumference of the circle are equal to half of the angle subtended by the same arc at the centre.

CONCLUSION: –

In the study of geometry, the relationship between angles and the circle is crucial. Several astronomical issues are directly related to the study of angles inside and outside the circle and their attributes.360° is the same as a circle. The circle can be divided into many smaller sections. An arc is a segment of the circle that is known according to its angle.

 Minor arcs (0°< v< 180°)

major arcs (180°< v <360°), and 

semicircles (v = 180°) are the three types of arcs.